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Methods of Constructing Index Numbers

Description: 76899.png 

 

Simple or Unweighted Index Numbers

In unweighted index numbers, the weights are not assigned to the variables, i.e., consideration is not given to the importance of each variable.

 

 

The following are different methods followed to calculate the unweighted index numbers.

  1. Simple aggregative method
     
    In this method of computing the price index, we express the total price of the commodity in a given year as a percentage of the total price of the commodity in the base year.
     
    Hence, Description: 67634.png
     
    Description: 67646.png  

 

Note: The index for the base year is always taken as 100.

 

Example
Commodities 1995 1996 1997
A 13 14 15.4
B 2 2.9 3.2
C 7 8 6.7
Total 22 24.9 25.3
Compute the simple aggregative index for 1996 and 1997 over 1995.
Solution
Simple aggregative index for 1996 over 1995 is Description: 67837.png 
Description: 67870.png 
Simple aggregative index for 1997 over 1995 is Description: 67897.png 
Description: 67917.png 
  1. Simple average of relatives method
     
    It can be derived using,
     
    Unweighted Arithmetic Mean Description: 67931.png
     
    Unweighted Geometric Mean  Description: 67939.png 
     
    Where,
     
    p01 = Index number of current year
     
    p1 = Total of current year’s price of all items
     
    p0 = Total of base year’s price of all items 

Example
Use the data given in Example 17.1 and find the simple average of relatives for the year 1997, taking 1995 as the base year.
Solution
Given:
Commodities p0 p1 Description: 67956.png
A 13 15.4 1.1846
B 2 3.2 1.60
C 7 6.7 .957
Total 22 25.3 3.7416


Simple average of relatives, 

 

Description: 67978.png 
Description: 67996.png 

Weighted Index Numbers

In this method, each variable is assigned a weight depending on its relative importance. Often the quantity or the volume of the commodity sold during the base year or some typical year may also be taken as the weights.

  1. Weighted aggregative index method
     
    a. Laspeyre’s index number:
     
    Description: 68018.png
     
    Here, weights assigned are base year quantities.
     

    Note: Laspeyre’s price index number is same as consumer price index number.

     
     
    b. Paasche’s index number:
     
    Description: 68177.png 
     
    Here, weights assigned are current year quantities.
     

    Note: Dorbish-Bowley index number is the A.M. of Laspeyre’s and Paasche’s index number.

     
     
    i.e.,Description: 68428.png
     
     
    c. Marshall–Edgeworth index number:
     
    Description: 68445.png 
     
    Here, the weights assigned are the average of base year and current year quantities.
     
     
    d. Fisher’s ideal index number:
     
    This formula is the geometric mean of Laspeyre’s and Paasche’s index numbers.
     
    Description: 68485.png 

Example
Compute the Laspeyre’s, Paasche’s, Marshall-Edgeworth and Fisher’s index number for the following data.
 
Solution
 
Description: 68523.png 
Description: 68529.png 
 
Marshall-Edgeworth index, 
Description: 68584.png 
 
Description: 68595.png 
 
Description: 68607.png 
  1. Weighted aggregative of relatives method
     
    The weighted arithmetic mean of price relatives using base year value weights is represented by
     
    Description: 68618.png 

Example
Calculate the weighted arithmetic mean index number from the following data.
Solution

The weighted arithmetic mean index number is
Description: 68653.png 

Quantity Index Number

When we want to measure and compare quantities, we resort to quantity index numbers. Quantity indices are used as indicators of the level of output in economy. These are calculated by adopting price as weights, i.e., by changing ‘p’ into ‘q’ and ‘q’ into ‘p’ in all the formulae for price index number. The various types of quantity indices are as follows:
  • Simple aggregate of quantities: Description: 68661.png
  • Simple average of quantity relatives: Description: 68679.png
  • Weighted aggregate quantity indices 
    1. Laspeyre’s quantity index number: 
       
      Description: 68686.png 
    2. Paasche’s index number: 
       
      Description: 68692.png 

Note: Dorbish-Bowley quantity index number is the A.M. of Laspeyre’s and Paasche’s index.

  1. Marshall–Edgeworth index number:
     
    Description: 68843.png 
  2. Fisher’s ideal index number:
     
    This formula is the geometric mean of Laspeyre’s and Paasche’s index numbers.
     
    Description: 68852.png 

Note: Base year weighted average of quantity relatives is given by

 

Example
Calculate
  1. Laspeyre’s quantity index number
  2. Paasche’s index number
  3. Dorbish–Bowley quantity index number
  4. Marshall–Edgeworth index number
  5. Fisher’s ideal index number for the given data.
Solution
 
The Laspeyre’s quantity index number isDescription: 69187.png 
 
The Paasche’s quantity index number is Description: 69196.png
 
The Dorbish–Bowley quantity index number is Description: 69204.png 
 
The Marshall–Edgeworth quantity index number is
 
Description: 69215.png 
 
The Fisher’s quantity index number is
 
Description: 69227.png 

Value Indices

Value is the product of price and quantity. Thus, a value index equals the total sum of the values of given year divided by the sum of the values of base year.

 

i.e., Description: 69237.png

Example
Calculate the value index number for the given data.
 
Solution
 
Description: 69312.png 





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