Paper 4: Fundamentals of Business Mathematics & Statistic

7.16 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Index Numbers

1. Unit Test - This test requires that the index number formulae should be independent of the units in which
   prices or quantities of various commodities are quoted. For example in a group of commodities, while
   the price of wheat might be in kgs., that of vegetable oil may be quoted in per liter & toilet soap may
   be per unit.
   Except for the simple (unweighted) aggregative index, all other formulae discussed above satisfy this
   test.


2. Time Reversal Test - The time reversal test is used to test whether a given method will work both
   backwards & forwards with respect to time. The test is that the formula should give the same ratio
   between one point of comparison & another no matter which of the two is taken as base.
   The time reversal test may be stated more precisely as follows—
   If the time subscripts of a price (or quantity) index number formula be interchanged, the resulting
   price (or quantity) formula should be reciprocal of the original formula.
   i.e. if p 0 represents price of year 2011 and p 1 represents price at year 2012 i.e.
   1
   0

p

p should be equal to 0 1

1

p p/

symbolically, the following relation should be satisfied

p 01 x p 10 = 1, Omitting the factor 100 from both the indices.

Where P 01 is index for current year ‘1’ based on base year ‘0’

pl0 is index for year ‘0’ based on year ‘1’.

The methods which satisfy the following test are:-

(1) Simple aggregate index

(2) Simple geometric mean of price relative

(3) Weighted geometric mean of price relative with fixed weights

(4) Kelly’s fixed weight formula

(5) Fisher’s ideal formula

(6) Marshall-Edgeworth formula

This test is not satisfied by Laspeyres’ method & the Paasche’s method as can be seen from below—

Σ ×Σ ≠

Σ Σ

1 0 0 1

0 0 1 1

p q pqpq p q 1(Laspeyres'Method)

Similarly when Paasche method is used—

Σ ×Σ ≠

Σ Σ

1 1 0 0

0 1 1 0

p q pqpq p q^1

On other hand applying Fisher’s formula

01 1 0 1 1

0 0 0 1

p pq pq

p q p q

= Σ × Σ

Σ Σ (Omitting the factor 100)

10 0 1 0 0

1 1 1 0

andp = ΣΣp q p qp q × ΣΣp q (Omitting the factor 100)