The economic purpose of the CPI is to convert nominal units of consumption (goods and services) into real units of consumption. I understand that the CPI is used for other many purposes, but that's the economic purpose. For example, say that there is one consumption good, corn, and corn is priced in denarii. We know that in 2010 50 billion denarii of corn was consumed while in 2018 100 billion denarii of corn was consumed. Was more corn consumed in 2010 or 2018? To answer this, you need to know the price of a kilogram of corn in denarii in 2010 and 2018. Say that $P_{2010}=25\:denarii/kilogram$ and $P_{2018}=100\:denarii / kilogram$ In this case consumption was 2 billion kilograms in 2010 and 1 billion kilograms in 2018.
When we get to multiple goods there is a complication because you can't add up or otherwise compare quantities of different goods. Therefore, it is convenient to pick a common numeraire. In the example, if we pick 2010 dollars as the numeraire then we see that consumption was 50 billion 2010 denarii in 2010 and 25 billion 2018 denarii in 2018. Both versions tell the same story, that real consumption was halved in 2018 relative to 2010.
This same logic holds when there are multiple goods. Let $t=0$ be the base (numeraire) year and $t=\tau$ be the year of comparison:
$$Real\: t=\tau\: Output\:in\:t=0\:prices =Expenditure_{good\:1,\tau}\cdot P_{1,0} / P_{1,\tau} + Expenditure_{good\:2,\tau} \cdot P_{2,0} / P_{2,\tau}\ldots + Expenditure_{good\:n,\tau} \cdot P_{n,0} / P_{n,\tau}$$
Recall that expenditures are prices times quantities, so expenditures divided by prices are quantities.
$$Real\: t=\tau\: Output\:in\:t=0\:prices =Q{1,\tau}\cdot P_{1,0} + Q{2,\tau} \cdot P_{2,0} \ldots + Q{n,\tau} \cdot P_{n,0}$$
While nominal output is:
$$Nominal\: t=\tau\: Output\:in\:t=\tau\:prices = Q{1,\tau}\cdot P_{1,\tau} + Q{2,\tau} \cdot P_{2,\tau} \ldots + Q{n,\tau} \cdot P_{n,\tau}$$
This lets us define the CPI:
$$\frac{Nominal\: t=\tau\: Output\:in\:t=\tau\:prices}{Real\: t=\tau\: Output\:in\:t=0\:prices} = CPI_{\tau}$$
That is, the relative change in the price level is the relative change in each prices weighted by the size of their expenditures.
For welfare purposes, it is important to understand that there are income and substitution effects from price changes. We can buy more of relatively cheaper goods and less of relatively expensive ones. A high price of a good we consume makes us effectively poorer. Both income and substitution can change our consumption bundles, and the Paasche, Laspeyres, and Fisher indices are attempts to deal with these issues. And there are other complications like changes in product availability, change in quality and features, and other measurement issues. But the overall idea is as above.
Therefore, if we are interested in making consumption comparisons across time, there really is only one way to do it, to weight the changes in prices by expenditures, and this means over-weighting the consumption of the rich because they consume more.
What happens if the weighted average change in the prices of the goods consumed by the poor is different than that of the rich? In this case the index is going to look more like the consumption of the rich than the poor. That, in and of itself, is not unfair, it is just an accounting result. This can become unfair when taxes and benefits are tied to indexed values. So, for example, if the goal is to guarantee everyone a minimum real income, and prices go up faster for the bundle consumed by the poor than the rich, the real value of a fixed benefit will go down.
This shows that the CPI is not the ideal index for all purposes. In my eyes, it is valuable to know (or at least to well approximate) the real consumption output. If the goal is instead to create an index for the benefits of the poor to preserve their real value then a separate index should be made for that purpose. The sensible goal of tying the index to its purpose leads to a proliferation of indices. In addition to the CPI, the US has the producer price index (AKA the PPI, a measure of intermediate good pricing), the personal consumption expenditures index (CPI is based on a survey of what households are buying; the PCE is based on surveys of what businesses are selling), and a consumer price index for the elderly. There probably are many others.
To my knowledge, there is no regularly updated government price index for the consumption of the poor. However Garner, Johnson, and Kokoski (1996) do develop such an index, and find it is very similar in trends to that of the more general CPI. Broda, Leibtag, and Weinstein (2009) and find "By examining scanner data on thousands of household purchases we find that the poor pay less —not more—for the goods they purchase." However, there does seem to be some contrary evidence more recently like Inflation May Hit the Poor Hardest (Casselman (2014)).
