INDEX NUMBERS

PRESENTED BY:

Mansi Doomra (130550034)Mehak Puri (130550035)Nidhi Aggarwal (130550039)Nikhil Khera (130550040)Shashank Kapoor (130550052)Vineet Kumar (130550062)

INTRODUCTION

An index number measures the relative change in price, quantity, value, or some other items of interest from one time period to another.

A simple index number measures the relative change in one or more than one variable.

DEFINITIONS

Index Numbers are a specialized type of averages. - M. Blair

Index Numbers are devices for measuring differences in the magnitude of a group of related values.

- Croxten and Cowden

An index number is a statistical measure designed to show changes in a variable or group of related variables with respect to time, geographical location or other characteristics. -Spigel

CHARACTERISTICS

Index numbers are specialized averages.

Index numbers measure the change in the level of a

phenomenon.

Index numbers measure the effect of changes over a period of time.

USES

• Index numbers give the knowledge as to what changes have occurred in the past.

HELPFUL IN PREDICTIONS

• By Index numbers relative changes occurring in the variables are determined, which simplifies the comparison of Data.

HELPFUL IN COMPARISONS

• Index numbers measures the changes taking place in the Business World and are useful in making a comparative study of the changes.

USEFUL IN BUSINESS

PROBLEMS RELATED TO INDEX NUMBER

Choice of the base period.

Choice of an average.

Purpose of index numbers.

Selection of commodities.

Data collection.

METHODS OF CONSTRUCTING INDEX NUMBERS

Ind

ex

Nu

mb

ers

Un-weighted

Simple

Aggregative

Simple Average of Price Relatives

Weighted

Weighted

Aggregative

Weighted Average of Price Relatives

SIMPLE AGGREGATIVE METHOD

In this method, sum of current year’s prices is divided by sum of base year’s prices and the quotient is multiplied by 100. Its formula is:

Where,P01= Index number of the current year. = Total of the current year’s price of all commodities. = Total of the base year’s price of all commodities.

1000

101

p

pP

1p

0p

EXAMPLE:From the data given below construct the index number for the year 2008 on the base year 2007 in Rajasthan state.

COMMODITIES UNITSPRICE (Rs)

2007PRICE (Rs)

2008

Sugar Quintal 2200 3200

Milk Quintal 18 20

Oil Litre 68 71

Wheat Quintal 900 1000

Clothing Meter 50 60

Solution:

COMMODITIES UNITSPRICE (Rs)

2007PRICE (Rs)

2008

Sugar Quintal 2200 3200

Milk Quintal 18 20

Oil Litre 68 71

Wheat Quintal 900 1000

Clothing Meter 50 60

32360 p 43511 p

Index Number for 2008:

45.341003236

4351100

0

101

p

pP

It means the prize in 2008 were 34.45% higher than the previous year.

In it, initially the price relatives of all the commodities are found out. To calculate price relatives, price of current year (p1) is divided by price of base year (p0) and then, the quotient is multiplied with 100.1. When ARITHMETIC MEAN is used:

2. When GEOMETRIC MEAN is used:N

pp

P

100

0

1

01

Where N is Numbers Of items.

N

pp

AntiP

100log

log 0

1

01

SIMPLE AVERAGE OF RELATIVES METHOD

3. When MEDIAN is used:

2

1 01

NofSizeP

SIMPLE AVERAGE OF RELATIVES METHOD

EXAMPLE:

From the data given below construct the index number for the year 2008 taking 2007 as base year by using arithmetic mean.

Commodities Price (2007) Price (2008)

P 6 10

Q 2 2

R 4 6

S 10 12

T 8 12

SOLUTION:

Index number using arithmetic mean

Commodities Price (2007) Price (2008) Price Relative

P 6 10 166.7

Q 12 2 16.67

R 4 6 150.0

S 10 12 120.0

T 8 12 150.0

1000

1 p

p

100

0

1

p

p= 603.37

63.1205

37.603100

0

1

01

N

pp

P

1p0p

WEIGHTED INDEX NUMBERS

When index numbers is constructed taking into consideration the importance of different commodities, then they are called weighted index numbers.There are two methods of contructing weighted index numbers.

1. Weighted Aggregative Index Numbers.2. Weighted Average of Price Relative Methods.

WEIGHTED AGGREGATIVE METHOD

In it, commodities are assigned weights on the basis of the quantities purchased. Different statisticians have used different methods of assigning weights, which are as follows:

Laspeyre’s method. Paasche’s method. Fisher’s ideal method. Dorbish and Bowley method. Marshall-Edgeworth’s method. Kelly’s method.

Laspeyre’s Method: This method was devised by Laspeyres in 1871. In this

method the weights are determined by quantities in the base.

10000

0101

qp

qpp

Paasche’s Method:

This method was devised by a German statistician Paasche in 1874. The weights of current year are used as base year in constructing the Paasche’s Index number.

10010

1101

qp

qpp

Dorbish & Bowleys Method:This method is a combination of Laspeyre’s and Paasche’s methods. If we find out the arithmetic average of Laspeyre’s and Paasche’s index we get the index suggested by Dorbish & Bowley.

Fisher’s Ideal Method:

Fisher’s ideal index number is the geometric mean of the Laspeyre’s and Paasche’s index numbers.

1002

10

11

00

01

01

qp

qp

qp

qp

p

10

11

00

01

01 qp

qp

qp

qpP 100

Marshall-Edgeworth Method:In this index the numerator consists of an aggregate of the current years price multiplied by the weights of both the base year as well as the current year.

Kelly’s Method:Kelly thinks that a ratio of aggregates with selected weights (not necessarily of base year or current year) gives the base index number.

1001000

110101

qpqp

qpqpp

1000

101

qp

qpp

Where q refers to the quantities of the year which is selected as the base. It may be any year, either base year or current year.

EXAMPLE

Given below are the price quantity data,with price quoted in Rs. per kg and production in qtls.Find:(1) Laspeyre’s Index (2) Paasche’s Index (3)Fisher Ideal Index.

ITEMS PRICE PRODUCTION PRICE PRODUCTION

BEEF 15 500 20 600

MUTTON 18 590 23 640

CHICKEN 22 450 24 500

2002 2007

Solution

ITEMS PRICE PRODUCTION

PRICE PRODUCTION

BEEF 15 500 20 600 10000 7500 12000 9000

MUTTON 18 590 23 640 13570 10620 14720 11520

CHICKEN 22 450 24 500 10800 9900 12000 11000

TOTAL 34370 28020 38720 31520

0p 0q 1q 1p

01qp 00qp 11qp 10qp

Solution

66.12210028020

34370100

00

0101

qp

qpp

2. Paasche’s Index:

84.12210031520

38720100

10

1101

qp

qpp

3. Fisher Ideal Index:

100 69.12210031520

38720

28020

34370

10

11

00

01

01 qp

qp

qp

qpP

1.Laspeyre’s index:

WEIGHTED AVERAGE OF PRICE RELATIVE

In weighted Average of Price relative, the price relatives for the current year are calculated on the basis of the base year price. These price relatives are multiplied by the respective weight of items. These products are added up and divided by the sum of weights.Weighted arithmetic mean of price relative is given by:

V

PVP01

1000

1 P

PPWhere:

P=Price relativeV=Value weights = 00qp

Quantity index numbers are designed to measure the change in physical quantity of goods over a given period. These index numbers represents increase or decrease in physical quantities of goods produce or sold. The method of construction of quantity index is same as that of price index. (1) Simple quantity index numbers (a) Simple Aggregative Method:

QUANTITY INDEX NUMBERS

1000

101

q

qQ

(b) Simple Average of Relative Method:

(i) Using A.M.

(ii) Using G.M.

N

qq

Q

100

0

1

01

N

qq

AntiQ

100log

log 0

1

01

(2) Weighted quantity index numbers

I. Weighted Aggregate Method

(a) Laspeyre’s quantity index no.:

(b) Paasche’s quantity index no.:

10000

0101

pq

pqQ

10010

1101

pq

pqQ

(c) Fisher’s quantity index numbers:

(d) Bowley’s quantity index:

(e) Marshall’s quantity index:

10010

11

00

01

01

pq

pq

pq

pqQ

1002

10

11

00

01

01

pq

pq

pq

pq

Q

1001000

110101

pqpq

pqpqQ

II. Weighted Average of Relative Method:

Where,

and

W

QWQ01

1000

1 q

qQ

00 pqW

VALUE INDEX NUMBERS

Value is the product of price and quantity. A simple ratio is equal to the value of the current year divided by the value of base year. If the ratio is multiplied by 100 we get the value index number.

10000

11

qp

qpV

TESTS OF ADEQUACY OF INDEX NUMBER FORMULAE

Various formulae can be used for the construction of index numbers but it is necessary to select an appropriate/suitable formula out of them. Prof. Fisher has given the following tests to select an appropriate formula:

TIME REVERSAL TEST (TRT) FACTOR REVERSAL TEST (FRT)

TIME REVERSAL TEST (TRT)

According to this test, if considering any year as a base year, some other year’s price index is computed and for another price index, time subscripts are reversed, then the both price indicies must be reciprocal to each other.TRT is satisfied when:

1 1

100110

01 PPorP

P

Where, P01 is price index for the year 1 with 0 as base and P10 is the price index for the year 0 with 1 as base.

FACTOR REVERSAL TEST (FRT)

Time reversal test permits interchange of price and quantities without giving inconsistent results, i.e. the two results multiplied together should give the true value ratio:

FRT is satisfied when:

Price Index x Quantity Index = Value Index OR

00

110101 qp

qpQP