where q0h is the vector of purchases for household h in the base year 0, x0h is its total expenditure in period 0, and is its share of total expenditures on good n in period 0. If, by contrast, we evaluate the national Laspeyres index using the aggregate bundle for all households, we would have
where the superscript A denotes “aggregate,” is the aggregate quantity defined as the sum of the individual quantities, X0 is aggregate expenditures on all goods and services, again the sum of the individual x0h, and is the share of aggregate expenditures on good n.
Both individual and aggregate Laspeyres indexes are weighted averages of the same price relatives, and the formulas (10) and (11) differ only in the weights. The aggregate index (11) uses the shares in the national budget, while the individual index (10) uses the shares in the household’s budget. The two sets of weights can be related to one another by noting that
so that the shares in the national budget are the weighted average of the shares in each household’s budget, where the weights are each household’s total expenditure as a share of national total expenditure. People who spend a lot count more in the national weights than do people who spend a little. Given (12), the individual and national Laspeyres indexes are related by
Equation (13) is the reason why the aggregate Laspeyres is referred to as a plutocratic index; each household’s individual Laspeyres price index is weighted by the total amount of money that it spends in period 0. This is in contrast to a democratic Laspeyres index in which each household’s index is averaged to obtain the national index
Note that the democratic and plutocratic Laspeyres indexes will coincide if everyone has the same income, or if everyone spends their money in the same proportions over the different goods, or if all the price relatives are equal.
Note finally that, if we combine (13) and (14), we can write