Business Mathematics and Statistics · Ch 9 — Applied Statistics (Time Series, Index Numbers, Statistical Quality Control)

Weighted Aggregate Methods: Laspeyres' and Paasche's Price Index

Because a simple index treats a commodity bought in huge quantity the same as one bought only rarely, real index numbers are weighted by quantity, so that commodities the public actually spends more on influence the index more. The two classical weighted-aggregate methods differ only in which year's quantities are used as weights.

Laspeyres' Price Index uses base-year quantities ( $q_0$ ) as weights — it asks: what would it cost, at current prices, to buy the same basket the base year actually bought?

P_{01}^{L} = \dfrac{\sum p_1 q_0}{\sum p_0 q_0} \times 100

Paasche's Price Index uses current-year quantities ( $q_1$ ) as weights instead — it asks: what does the current basket cost now, compared with what it would have cost at base-year prices?

P_{01}^{P} = \dfrac{\sum p_1 q_1}{\sum p_0 q_1} \times 100

Worked Example. Price and quantity data for three commodities:

Commodity	$p_0$	$q_0$	$p_1$	$q_1$	$p_0q_0$	$p_1q_0$	$p_0q_1$	$p_1q_1$
X	5	10	6	12	50	60	60	72
Y	8	5	9	6	40	45	48	54
Z	10	4	12	5	40	48	50	60
Total					130	153	158	186

Laspeyres' Price Index:

P_{01}^{L} = \dfrac{153}{130}\times 100 = 117.69

Paasche's Price Index:

$P_{01}^{P} = \dfrac{186}{158}\times 100 = 117.72$ …

Definition 15Laspeyres' Price Index

A weighted aggregate price index that uses base-year quantities ( $q_0$ ) as weights: $P_{01}^{L}=\dfrac{\sum p_1 q_0}{\su …

Definition 16Paasche's Price Index

A weighted aggregate price index that uses current-year quantities ( $q_1$ ) as weights: $P_{01}^{P}=\dfrac{\sum p_1 q_1}{\su …

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