Weighted Index Numbers: Laspeyres, Paasche and Fisher's Ideal

In real life, a family spends far more on food than on stationery, so a price rise in food should influence the overall index far more than an equal percentage rise in stationery. Weighted index numbers fix this by attaching a weight — usually the quantity consumed — to each commodity.

(A) Laspeyres' Price Index uses base-year quantities ($q_0$) as weights:

It answers: "what would it cost, at current prices, to buy the base year's basket?" Because the basket is fixed at the (older) base-year pattern, Laspeyres tends to slightly overstate the price rise, since it ignores that consumers shift away from goods that have become relatively more expensive.

(B) Paasche's Price Index uses current-year quantities ($q_1$) as weights:

It answers: "what would the current basket have cost at base-year prices, versus what it actually costs now?" Because it uses the (newer) current-year basket throughout, Paasche tends to slightly understate the price rise.

(C) Fisher's Ideal Index is the geometric mean of the Laspeyres and Paasche indices, and is named "ideal" because it satisfies both tests of adequacy discussed in the next section:

Weighted Average of Price Relatives is an equivalent route to the same idea: weight each commodity's price relative by its base-year value weight $w = p_0 q_0$:

$P_{01} = \dfrac{\Sigma\left(w \times \dfrac{p_1}{p_0}\times 100\right)}{\Sigma w}$ …