The CPI for use in calculating the rate of inflation is based off of Laspeyres' index, which is essentially the ratio of dot products of the price vector and a quantity vector. What kind of analysis would we be unable to do if we constructed a price index that is just the ratio of the sum of new prices to the sum of old prices?
I ask this because welfare analysis as far as compensating/equivalent variations are concerned looks only at the change in prices.
Of course, looking at prices alone will not allow us to infer substitution effects and so such measure would be a biased measure of welfare. But so too is there such bias if, as with Laspeyres' index, we hold the quantity vector constant. So, the question boils down to: is the Laspeyres' index useful only in how we could couple this with Paasche's to obtain Fisher's ideal index?