Another way of tackling this question is to ask what kind of household has a consumption pattern that matches the CPI? The answer is as follows:

It is natural to ask then what is the household better represented by the plutocratic CPI. Muellbauer (1974) searched for the household whose budget shares were closest to the ... aggregate weights in the UK CPI, and found it to be at the 71st percentile in the household expenditures distribution. For the US in 1990, Deaton (1998) estimates that this consumer occupies the 75th percentile. Thus, the 'representative' consumer embedded ... is inclined towards upper-expenditure households.

Ley cites a 1987 study by Kokoski that estimates the plutocratic gap at -0.1 to -0.3 percentage points per year over the 1972-80 period. In words, this means that CPI using democratic weights experienced between 0.1 to 0.3 percentage points greater inflation than the reported CPI inflation rate.

More recently, Kokoski (2003) has updated her analysis (a related version published in Monthly Labor Review in 2000, see here). She summarizes her paper thus:

This paper provides an empirical analysis of the differences between the plutocratic and democratic price indices, using data from the Consumer Expenditure Survey and the CPI for the periods 1987-1997, and for simulated price change scenarios. The results show that there is very little difference between the two types of index, and that one index need not always exceed the other. In the simulated scenarios, even the extreme cases where prices changed only for expenditure-inelastic goods and services, the difference between the democratic and plutocratic indices was only about one point for every ten percent increase in the relative prices of these goods.

Can we extend these results to the present time? It's not clear. There is the conjecture that, with lower income households having a basket skewed toward food and gasoline, the plutocratic gap would be wide, particularly over the last couple years. While that conjecture makes sense to me, I'd say that answer is actually not clear. The reason I say that is because of a recent paper by Broda and Romalis, who note that because of Chinese imports, lower income households have actually benefitted from globalization to a much greater degree than typically thought exactly because they have consumption patters skewed toward goods that have decreased in price over the past decade. From Broda and Romalis's paper:

… we find that inflation for households in the lowest tenth percentile of income has been 6 percentage points smaller than inflation for the upper tenth percentile over this period. The lower inflation at low income levels can be explained by three factors: 1) The poor consume a higher share of non-durable goods -- whose prices have fallen relative to services over this period; 2) the prices of the set of non-durable goods consumed by the poor has fallen relative to that of the rich; and 3) a higher proportion of the new goods are purchased by the poor. We examine the role played by Chinese exports in explaining the lower inflation of the poor.