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Index Number

Index numbers are special type of averages to compare two or more year or series or data , in which one year or series is base and another year or series is considered as current . Index number give the change of group variable having different measuring unit.

Base year is denoted by o and current year is denote by 1 .

The index number of base year will always be equal to 100.

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Price Relative

It is relative change of current year to base year, it is denoted by i

Formula : i =

P₁/P₀

x 100

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Some Important Formulas :

(i)     Simple Aggregate Price Index Number (P₀₁) =

∑P₁/∑P₀

x 100

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(ii)   Simple Price Relative Index Number (P₀₁) =

∑I/n

   where I = is price relative and n is number of commodity.

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(iii)    Weighted Price Relative Method (P₀₁) =

∑WI/∑W

where I = price relative and W = weight.

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(iv)   Weighted Aggregate Price Index Number (P₀₁) =

∑P₁W/∑P₀W

x 100

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Marshall-Edgeworth Price Index Number

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Kelly Price Index Number



where weight is equal to q(quantity), either current year or base year.

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Dorbish Bowley Price Index Number

It is arithematic mean of laspeyers and paasches price index number



or

P₀₁ = (L + P)/2

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Fisher's Price Index Number

It is geometric mean of laspeyers and paasches price index number

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Test of Adequacy (Index Numbers)

There are four tests:

(a) Unit Test
(b) Time Reversal Test
(c) Factor Reversal Test
(d) Circular Test

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(a) Unit Test: In this the formula should be independent of the unit in which or for which prices and quantities are equal are quoted.
All formula of Index Number satisfy this test except simple unweighted aggregative Index Number.

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(b) Time Reversal Test: This test was suggested by Prof. Irving Fisher.When the data for only two years are treated and will work in both ways in time, forward and backward. There are two ratios Such that, the current on the base and the base on the current should multiply into unity.

P₀₁ x P₁₀ = 1 or P₀₁ = 1/P₁₀

Where P₀₁ is the index for time 1 on 0 and P₁₀ is the index for time 0 on 1. Laspeyer's and Paasche's method do not satisfy this test. Fisher's, M-E, Walsch's, Kelly's, Simple aggregate index number satisfy this test.

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(c) Factor Reversal Test: This test is suggested by Prof. Irving Fisher. According to this test the product of price index and the quantity index must be equal to the value index ,
i.e. ∑(P₁Q₁)/∑(P₀Q₀)

Symbolically :P₀₁ x Q₀₁ = V₀₁
Only Fisher's index number satisfies this test and no other index number satisfy this test.

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(d) Cirular Test: This test is an extension of time reversal test(It is a generalisation of the time reversal test). The test of shiftability of base is called the circular test.

P₀₁ x P₁₂ x P₂₃ ------ P_n-1,n x P_n0 = 1.

Laspeyer's, Paasche's and Fisher's index number do not satisfy test. The simple aggregative method and the fixed weight aggregative method satisfy this test.

Base Shifting

Shifting of base period or reference period of the index is known as base shifting .

This can be done with the following relation as .

Shifting price index =

original price index/price index for new base year

x 100

Example : The following are the index numbers of a commodity taking 2000 as the base .

Years :	2000	2001	2002	2003	2004
Index number :	100	110	140	150	180

Find the index number by changing the base to 2003

Solution .

Years :	Old index number (Base year 2000)	New index number (Base year 1994)
2000	100	original price index/price index for new base year x 100 = 100/150 x 100 = 66.67
2001	110	110/150 x 100 = 73.33
2002	140	140/150 x 100 = 93.33
2003	150	150/150 x 100 = 100
2004	180	180/150 x 100 = 120

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Splicing of Series

Splicing of Series is the process to combine two or more series (or sets) of overlapping index number .

There are two types of splicing

(a) . Backward Splicing .

(b) . Forward Splicing .

Example

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Years :	2000	2001	2002	2003	2004	2005	2006	2007	2008
Index number :2000 = 100	100	120	130	140	150
Index number : 2004 = 100					100	110	130	140	160

Solution (i) . Backward Splicing

Years :	2000	2001	2002	2003	2004	2005	2006	2007	2008
Index number :2000 = 100	100	120	130	140	150
Index number : 2004 = 100					100	110	130	140	160
Backward splicing	100 x 100/150 = 66.67	120x100/150 = 80	130 x 100/150 = 86.67	140 x 100/150 = 93.33	100	110	130	140	160

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Solution (ii) . Forward Splicing

Years :	2000	2001	2002	2003	2004	2005	2006	2007	2008
Index number :2000 = 100	100	120	130	140	150
Index number : 2004 = 100					100	110	130	140	150
Forward splicing	100	120	130	140	150	150x110/100 = 165	150x130/100 = 195	150x140/100 = 210	150x150/100 = 225

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