Construction of a Price Index — Weighted Methods

A student's family does not spend equally on rice and on matchsticks — a genuinely realistic price index must give more IMPORTANCE (weight) to commodities that account for a larger share of actual spending. Weighted methods fix this by attaching a weight, usually the QUANTITY consumed/purchased, to each commodity.

1. Weighted Aggregative Methods. These weight the price totals directly using quantities as weights, but different formulae choose the quantity weights from a different year: Laspeyres' Method uses BASE-YEAR quantities ($q_0$) as weights for both the numerator and the denominator:

   $L = \frac{\sum p_1 q_0}{\sum p_0 q_0}\times 100$

   Because the SAME base-year basket ($q_0$) is priced at both years' prices, Laspeyres' index answers a clean question: "how much MORE would the base year's basket cost today?" Its main limitation is that consumption patterns genuinely shift over time (people buy less of a good that has become relatively expensive), so pricing an outdated, unchanged basket can overstate the real rise in the cost of living. Paasche's Method uses CURRENT-YEAR quantities ($q_1$) as weights for both the numerator and the denominator:

   $P = \frac{\sum p_1 q_1}{\sum p_0 q_1}\times 100$

   Paasche's index answers a different clean question: "how much would TODAY's basket have cost in the base year?" Because it uses the CURRENT basket, it automatically reflects any shift consumers have already made away from goods that got relatively costlier — but it also means the basket itself keeps changing every year, making the Paasche index harder to compare across many successive years. Fisher's Ideal Method is the GEOMETRIC MEAN of the Laspeyres and Paasche indices:

   $F = \sqrt{L\times P} = \sqrt{\frac{\sum p_1q_0}{\sum p_0q_0}\times\frac{\sum p_1q_1}{\sum p_0q_1}}\times 100$

   By combining a base-year-weighted view and a current-year-weighted view, Fisher's index balances out the specific bias each one carries alone — this is exactly why it is called "ideal" (Section 5 makes this precise using two formal tests).
2. Weighted Average of Price Relatives Method. Here, instead of weighting price TOTALS, each commodity's own price RELATIVE is weighted, usually by its base-year value $w=p_0q_0$: …