Weighted Index Numbers
When all commodities are not of equal importance, we assign weight to each commodity relative to its importance and the index number computed from these weights is called a weighted index number.
Laspeyre’s Index Number
In this index number the base year quantities are used as weights, so it also called the base year weighted index.
\[{P_{on}} = \frac{{\sum {P_n}{q_o}}}{{\sum {P_o}{q_o}}} \times 100\]
Paasche’s Index Number
In this index number the current (given) year quantities are used as weights, so it is also called the current year weighted index.
\[{P_{on}} = \frac{{\sum {P_n}{q_n}}}{{\sum {P_o}{q_n}}} \times 100\]
Fisher’s Ideal Index Number
The geometric mean of Laspeyre’s and Paasche’s index numbers is known as Fisher’s ideal index number. It is called ideal because it satisfies the time reversal and factor reversal test.
\[\begin{gathered} {P_{on}} = \sqrt {{\text{Laspeyre’s Index }} \times {\text{ Paashe’s Index}}} \\ {P_{on}} = \sqrt {\frac{{\sum {P_n}{q_o}}}{{\sum {P_o}{q_o}}} \times \frac{{\sum {P_n}{q_n}}}{{\sum {P_o}{q_n}}}} \times 100 \\ \end{gathered}\]
Marshal-Edgeworth Index Number
In this index number the average of the base year and current year quantities are used as weights. This index number was proposed by two English economists, Marshal and Edgeworth.
\[\begin{gathered} {P_{on}} = \left( {\frac{{\sum {P_n}{q_o} + \sum {P_n}{q_n}}}{{\sum {P_o}{q_o} + \sum {P_o}{q_n}}}} \right) \times 100 \\ {P_{on}} = \frac{{\sum {P_n}\left( {{q_o} + {q_n}} \right)}}{{\sum {P_o}\left( {{q_o} + {q_n}} \right)}} \times 100 \\ \end{gathered}\]
Example:
Compute the weighted aggregative price index numbers for $$1981$$ with $$1980$$ as the base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number (4) Marshal-Edgeworth Index Number.
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Commodity
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Prices
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Quantities
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||
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$$1980$$
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$$1981$$
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$$1980$$
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$$1981$$
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|
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$$A$$
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$$10$$
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$$12$$
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$$20$$
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$$22$$
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$$B$$
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$$8$$
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$$8$$
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$$16$$
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$$18$$
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$$C$$
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$$5$$
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$$6$$
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$$10$$
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$$11$$
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$$D$$
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$$4$$
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$$4$$
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$$7$$
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$$8$$
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Solution:
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Commodity
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Prices
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Quantity
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$${P_1}{q_o}$$
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$${P_o}{q_o}$$
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$${P_1}{q_1}$$
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$${P_o}{q_1}$$
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$$1980$$
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$$1981$$
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$$1980$$
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$$1981$$
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|||||
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$${P_o}$$
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$${P_1}$$
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$${q_o}$$
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$${q_1}$$
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|||||
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$$A$$
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$$10$$
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$$12$$
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$$20$$
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$$22$$
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$$240$$
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$$200$$
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$$264$$
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$$220$$
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$$B$$
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$$8$$
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$$8$$
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$$16$$
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$$18$$
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$$128$$
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$$128$$
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$$144$$
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$$144$$
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$$C$$
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$$5$$
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$$6$$
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$$10$$
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$$11$$
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$$60$$
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$$50$$
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$$66$$
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$$55$$
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$$D$$
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$$4$$
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$$4$$
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$$7$$
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$$8$$
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$$28$$
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$$28$$
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$$32$$
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$$32$$
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$$\begin{gathered} \sum {P_1}{q_o} \\ = 456 \\ \end{gathered} $$
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$$\begin{gathered} \sum {P_o}{q_o} \\ = 406 \\ \end{gathered} $$
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$$\begin{gathered} \sum {P_1}{q_1} \\ = 506 \\ \end{gathered} $$
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$$\begin{gathered} \sum {P_o}{q_1} \\ = 451 \\ \end{gathered} $$
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||||
Laspeyre’s Index Number
\[{P_{on}} = \frac{{\sum {P_n}{q_o}}}{{\sum {P_o}{q_o}}} \times 100 = \frac{{456}}{{406}} \times 100 = 112.32\]
Paashe’s Index Number
\[{P_{on}} = \frac{{\sum {P_n}{q_n}}}{{\sum {P_o}{q_n}}} \times 100 = \frac{{506}}{{451}} \times 100 = 112.20\]
Fisher’s Ideal Index Number
\[\begin{gathered} {P_{on}} = \sqrt {{\text{Laspeyre’s Index }} \times {\text{ Paashe’s Index}}} \\ {P_{on}} = \sqrt {{\text{112}}{\text{.32}} \times {\text{112}}{\text{.20}}} = 112.26 \\ \end{gathered} \]
Marshal-Edgeworth Index Number
\[\begin{gathered} {P_{on}} = \left( {\frac{{\sum {P_n}{q_o} + \sum {P_n}{q_n}}}{{\sum {P_o}{q_o} + \sum {P_o}{q_n}}}} \right) \times 100 \\ {P_{on}} = \left( {\frac{{456 + 506}}{{406 + 451}}} \right) \times 100 = \frac{{962}}{{856}} \times 100 = 112.38 \\ \end{gathered} \]