Weighted Index Numbers

When all commodities are not of equal importance. We assign weight to each commodity relative to its importance and index number computed from these weights is called weighted index numbers.

Laspeyre’s Index Number:

In this index number the base year quantities are used as weights, so it also called 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 current year weighted index.

{P_{on}} = \frac{{\sum {P_n}{q_n}}}{{\sum {P_o}{q_n}}} \times 100

Fisher’s Ideal Index Number:

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 is 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 1980as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number (4) Marshal Edgeworth Index Number.

Commodity
Prices
Quantities
1980
1981
1980
1981
A
10
12
20
22
B
8
8
16
18
C
5
6
10
11
D
4
4
7
8

Solution:
           

Commodity
Prices
Quantity
{P_1}{q_o}
{P_o}{q_o}
{P_1}{q_1}
{P_o}{q_1}
1980
1981
1980
1981
{P_o}
{P_1}
{q_o}
{q_1}
A
10
12
20
22
240
200
264
220
B
8
8
16
18
128
128
144
144
C
5
6
10
11
60
50
66
55
D
4
4
7
8
28
28
32
32
 
\begin{gathered} \sum {P_1}{q_o} \\ = 456 \\ \end{gathered}
\begin{gathered} \sum {P_o}{q_o} \\ = 406 \\ \end{gathered}
\begin{gathered} \sum {P_1}{q_1} \\ = 506 \\ \end{gathered}
\begin{gathered} \sum {P_o}{q_1} \\ = 451 \\ \end{gathered}

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}