9c462Index Number

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Module - 4

INDEX NUMBERS

Contents:

4.1 Introduction4.2 Problems Involved in the Construction of Index Numbers

4.3 Types of index numbers

4.3. 1 Price index,

4.3.2 Quantity index

4.3.3 Value index.

4.3.4 Special purpose

4.4 Methods of construction

4.4.1 Unweighted Indexes

4.4.2 Weighted Indexes

4.5 Base Shifting

4.6 Splicing of Indexes

4.7 Deflating of Index Number

4.8 Uses of Index Numbers

4.9 Limitations of Index Numbers


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Module 4

INDEX NUMBERS

4.1. INTRODUCTION

Index numbers are the indicators, which reflect changes over a specified period of time in

1. Prices of different commodities

2. Industrial production

3. Sales

4. Imports and exports

5. Cost of living, etc.

These indicators are of paramount importance to the management personnel or any government organisation or

industrial concern for the purpose of reviewing position and planning action, if necessary; and in the formulation of

executive decisions. They reflect the pulse of an economy and serve as indicators of inflationary or deflationary

tendencies. Just as in Physics and Chemistry barometer measures atmospheric pressure or pressure of gases,

so in Economics index numbers measure the pressure of economic behaviour and are rightly termed as

'economic barometers' or 'barometers of economic activity' since a look at some of the important indices like

index numbers of wholesale prices, industrial production, agricultural production, etc., gives a fairly good idea as

to what is happening to the economy of a country.

Definition:

"Index numbers are statistical devices designed to measure the relative change in the level of a phenomenon

(variable or a group of variables) with respect to time, geographical location or other characteristics such as

income, profession, etc." In other words, these are the numbers which express the value of a variable at any

given date called the 'given period' as a percentage of the value of that variable at some standard date called the

'base period'. The variable may be:

(i) The price of a particular commodity, e.g., silver, iron, etc., or a group of commodities like consumer goods,

foodstuffs, etc;

(ii) The volume of trade, exports and imports, agricultural or industrial production, sales in a departmental store,

etc;

(iii) The national income of a country or cost of living of persons belonging to particular income group/profession,

etc.


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For example, suppose we want to measure the general changes in the price level of consumer goods, i.e.,

goods or commodities consumed by the people belonging to a particular section of society, say, low-income

group or middleincome group or labour class, and so on. Obviously, these changes are not directly measurable

as the price quotations of various commodities are available in different units, e.g., wheat and- sugar in Rs. per

quintal, milk, petrol and kerosene in Rs. per litre, cloth in Rs. per metre, etc. Further, the prices of some of the

commodities may be increasing while those of others may be decreasing during the two periods and the rates of

increase or decrease may be different for different commodities. Index number is a statistical device which

enables us to arrive at a single representative figure which gives the general level of the price of the

phenomenon (commodities) in an extensive group. According to Wheldon, "An index number is a device which

shows by its variation the changes in a magnitude which is not capable of accurate measurement in itself or of

direct valuation in practice."

Edgeworth gave the classical definition of index numbers as follows: "Index number shows by its variations the

changes in a magnitude which is not susceptible either of accurate measurement in itself or of direct variation inpractice."

4.2. Problems Involved in the Construction of Index Numbers

The methods of construction of index numbers warrant a careful study of the following problems:

1. The purpose of Index Number. An index number which is properly designed for a purpose can be most useful

and powerful tool otherwise it can be equally misleading and dangerous. Thus the first and foremost problem is

to determine the purpose of index number without which it is not possible to follow the steps in its construction.

Moreover, precise statement of the purpose usually settles some related problems, e.g., if the purpose of Index

number is to measure the changes in the production of steel, (say), the problem of selection of items

(commodities is automatically settled.

2. Selection of Commodities. Having defined the purpose of index numbers, select only those commodities

which are relevant to the index. For example, if the purpose of an index: is to measure the cost of living of low

income group (poor families) we should select only those commodities or items which are consumed/utilised by

persons belonging to this group and due care should be taken not to include the goods/services which are

ordinarily consumed by middle-income or high-income group For such an index, selection of commodities like

cosmetics and other luxury goods like scooters, cars, refrigerators, television sets, etc., will be absolutely

useless.

The best solution to the problem of selection of items for any index is (i) to split the whole (relevant) group of

commodities into various homogeneous subgroups like cereals, milk and milk products, clothing, iron and steel,

electrical appliances and fuel, etc., so that the price movement of various commodities within any subgroup


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follows almost the same pattern, and (ii) to select an adequate number of representative items from each sub-

group.

Remark. It should be borne in mind that for the index number the same grade/quality of the commodities, say,

wheat, rice, etc., is included at different times. In order to avoid confusion due to time-lag about the qualities, it is

desirable to include, as far as possible, only standardized or graded items.

3. Data for Index Numbers. The data, usually the set of prices and of quantities consumed of the selected

commodities for different periods, places, etc., constitute the raw material for the construction of index numbers.

The data should be collected from reliable sources such as standard trade journals, official publications,

periodical special reports from the procedures, exporters, etc., or through field agency. The principles of data

collection, viz., accuracy, comparability, sample representatives and adequacy should be borne in mind. In any

case the data should strictly pertain to what is being measured. For example, for the construction of retail price

index numbers, the price quotations for an adequate number of commodities (used by a particular group of

people for whom the index is intended) should be obtained from superbazars, fair price shops, departmentalstores, etc., and not from wholesale dealer.

4. Selection of Base Period. The period with which the comparisons of relative changes in the level of a

phenomenon are made is termed as 'base period' and the index for this period is always taken as 100. The

following are basic criteria for the choice of the base period.

(i) The 'base period' must be a 'normal period.' i.e., a period free from all sorts of abnormalities or chance

fluctuations such as economic boom or depression, labour strikes, wars, floods, earthquakes etc. If the base

period be taken as a period of economic instability or depression in which the prices of various commodities and

goods, due to their scarcity, have been abnormally high then the comparison of price relatives in any given year

will not be of much practical utility.

(ii) The base period should not be too distant from the given period. Since index numbers are essential tools in

business planning and in formulation of executive decisions, the base period should not be too far back in the

past relative to the given period because due to dynamic pace of events these days, distant base period is likely

to be entirely different from the given period. Moreover. if the base year is shifted far away from the given

period, it is possible that the pattern of consumption of commodities may change appreciably. For example, for

deciding about grant of D.A. (dearness allowance) increment to government personnel, the prices should be

compared with the period when last D.A. was announced or granted.

5. Type of Average to be used. Since index numbers are specialised averages, a judicious choice of average to

be used in their construction is of great importance. Usually the following averages are used:


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(i) Arithmetic Mean (A.M.): simple or weighted,

(ii) Geometric Mean (G.M.): simple or weighted, (iii) Median.

Median, though easiest to calculate of all the three, completely ignores the extreme observations while arithmetic

mean, though easy to calculate, is unduly affected by extreme observations. Moreover, as we shall see later,

neither arithmetic mean nor median are reversible and hence do not reflect typical movements of prices or

quantities. Since in the construction of index numbers we deal with ratio or relative changes and since geometric

mean.

(i) Gives equal weights to equal ratios of change,

(ii) Does not give undue weightage to extreme observations, and

(iii) G.M. based indices are reversible, from theoretical considerations G.M. is the most appropriate average to be

used. But in spite of its theoretical claim, in practice G.M. is not used as often as A.M. because of its

computational difficulties. However, in the interest of greater accuracy and precision, G.M. should berecommended.

6. Selection of appropriate weights.. Generally, various items commodities, say, wheat, rice, kerosene, clothing,

etc., included in the index are not of equal importance, proper weights should be attached to them to take into

account their relative importance. Thus there are two types of indices:

(i) Unweighted Indices in which no specific weights are attached to various Commodities; and

(ii) Weighted Indices in which appropriate weights are assigned to various items.

Strictly speaking, unweighted indices can be interpreted as weighted indices, the corresponding weight for each

commodity being unity. The question of allocating suitable weights is of fundamental importance but at the same

time quite difficult also. The various forms of weights usually used in practice are discussed below in the various

formulae for the construction of index numbers.

7. Choice of Formulae. The availability of information regarding the prices and quantities of the selected

commodities serves as a prerequisite for the construction of index numbers which involves the following

steps:Notations. Let

Pij denote the price of jth commodity in the ith year.

qij denote the quantity of the jth commodity consumed in the ith year.

Vij = Pij X qij denote the value of jth commodity in the ith year.


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where j = 0, 1, 2, ..., n and i = 0, 1, 2, ..., k refer to the various situations to be compared, which we have referred

to as years; '0' serving as the base year and 'i' as the given year.

In the following sequences, the summation is taken over j from 1 to n, unless otherwise stated. Thus, we will

write

n pij = pijj=1n qij = qijj=1

Which are the ith year price and quantity respectively? In particular, p 0j and qOj refer to base year price and

base year quantity respectively.

4.3 Types of index numbers

1. Price index,

2. Quantity index

3. Value index.

4. Special purpose

4.3.1 The price index is a summary measure that combines the price changes for a group of items, using weights

to give each item its appropriate importance.

Example:

The consumer price index (CPI) measures the combined effect of price changes in many goods and services

purchased by households with 1994 as base year (as last revised).

4.3.2 The quantity index is a summary measure of relative changes over time in the quantities or volume of

some measurable characteristic such as production, sales, inventories, or consumption of a specific commodity or

group of commodities.

Example: The volume of production index (VoPI) monitors the changes in the quantity of a commodity produced

or manufactured by the establishments.

4.3.3 The value index are summary measures of relative changes over time in the value (price of commodity x

quantity consumed) of a commodity or group of commodities.A value index measures changes in both the price

and quantities involved.

A value index, such as the index of department store sales, needs the original base-year prices, the original base-

year quantities, the present-year prices, and the present year quantities for its construction.


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Value index = =_ pij qij_ X 100

poj qoj

4.4. Methods of construction

Unweighted Indexes

Simple Average of the Price Indexes

Simple Aggregate Index

Weighted Indexes

Laspeyres Price Index

Paasche Price Index

Dorbish and Bowleys Index

Fishers Ideal Index

Marshall-Edgeworth Index

4.4.1 Unweighted Indexes

Simple Average - Example


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Simple Aggregate Index Example

4.4.2 Weighted Indexes

There are several types of indices defined, among them those listed in the following table.

S.No. Index Formula

1 Laspeyres Index pij. q0j X 100 p0j. q0j

2 Paasches Index pij. qij X 100 p0j. qij

3 Bowley Index1/2[ pij. q0j + pij. qij ] X 100

p0j. qij p0j. q0j


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4 Fisher Index (Laspeyres Index* Paasches Index)1/2

5 Marshall-Edgeworth Index pij. (q0j + qij ) X 100 p0j(q0j + qij )

Laspeyres Price Index : Laspeyres Price Index uses quantities as weights. It is based on the assumption that the

amount purchased by a typical family does not change from the base year amount. Thus the base year quantitiesare used as weights.

The Laspeyres index is convenient to use on a continuing basis because the weights remain fixed from one

period to the next.

Laspeyres Index

PLa =_ pij qoj_ X 100

01 poj qoj

Paasches price index :Paasches price index uses quantities as weights. It requires a new set of weights for

every period since the ithperiod quantities are used as weights. For this reason, Laspeyres is used far more often

than the Paasche index for measuring price changes.

In general, Laspeyres index tends toward an upward bias, while Paasche index tends to underestimate the true

changes in price.

Paasches Index

PPa = _ pij qij_ X 100

01 poj qij

Dorbish and Bowleys price Index :Dorbish and Bowleys price Index is a combination of Laspeyres and

Paasches methods. If we find out the arithmetic average of Laspeyres and Paasches index we get the

index suggested by Drobish & Bowley.

PDB = (PLa + PPa )

01 2

Fishers ideal index: Fishers ideal index is another compromise between the use of a constant base year and

current period weights. It combines the information summarized by the Laspeyres and Paasche index by

averaging them using the geometric mean.

Fishers index shares the disadvantage of using the Paasche index which requires the computation of a new

set of weight for each new period. Thus, it is still not so ideal from a practical point of view.


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Fishers Ideal Index

PF = (PLa X PPa ) 1/2

0i 0i 0i

The Fishers ideal index number has the following subtle mathematical properties thats why it is called ideal:

1) Time reversal property If the time subscripts of the index number formula are interchanged, the resulting

index is the reciprocal of the original formula.

2) Factor reversal property If the x and w factors in the index formula are interchanged so that the original value

of the variable are now the weights and the original weights are now the value of the variable, the

product of the two indexes should give the true value ratio.

Edgeworth-Marshall Price index :The Edgeworth-Marshall index is a practical compromise between the use ofa constant base year weights and ith period weights by using using the average of the two weights.

Just like Paasches index, it also requires a new set of weights for each period which makes it more difficult to use

in maintaining a series of index numbers compared to Laspeyres

PDB = pij. (q0j + qij ) X 100

oi p0j(q0j + qij )

Illustration: The table below relates to the daily pay of the wage earners on a company's pay roll

April 1978 April 1983

Number Total pay Number Total pay

(00 Rs.) (00 Rs.)

Men aged 21 and over 350 7.14 300 14

Women aged 18 and over 400 4 1200 6.67

Youths and boys 150 3 100 5.6

Girls 100 2.5 400 3.85

1000 16.64 2000 30.12

Construct an index of daily earnings based on 1978 as base showing the rise of earnings for all employees

as one figure.


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Solution. Regarding number of wage earners as quantities and the weekly wages per labourer as prices we

are given the figure qo and poqo for 1978 and the values q1 and plql for 1983. The following table can be easily

complete

qo po q1 p1 poqo p1q0 poq1 p1q1

350 7.14 300 14 2,499 4,900 2,142 4,200

400 4 1,200 6.67 1,600 2,668 4,800 8,004

150 3 100 5.6 450 840 300 560

100 2.5 400 3.85 250 385 1,000 1,5401000 16.64 2000 30.12 4,799 8,793 8,242 14,304

Laspeyres Index

PLa =_ pij qoj_X 100 = 8793 X 100 =183.22

0i poj qoj 4799

Paasches Index

PPa =_ pij qij_X 100 = 14300 X 100 =173.55

0i poj qij 8242

Dorbish and Bowleys Index

PDB = (PLa + PPa ) = 178.38

0i 2

Fisher Index

PF = (PLa X PPa ) 1/2 = (183 X 173.5)1/2 = 178.32

0i 0i 0i

4.5 Base Shifting of Index Numbers.

Base shifting means the changing of the given base period (year) of a series of index numbers and recasting

them into a new series based on some recent new base period. This step is quite often necessary under the

following situations

(i) When the base year is too old or too distant from the current period to make meaningful and valid

comparisons.

(ii) If we want to compare series of index numbers with different base periods, to make quick and valid

comparisons both the series must be expressed 'with a common base period.

Base shifting requires the recomputation of the entire series of the index numbers with the new base. However,


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this is a very difficult and. time consuming job. A relatively much simple, though approximate method consists in

taking the index number of the new base year as 100 and then expressing the given series of index numbers as a

percentage of the index number of the time period selected as the new base year. Thus, the series of index

numbers, recast with a new base is obtained by the formula

Recast Index No. of any year

=. Old Index No. of the year X 100

Index No. of new base year

=. 100 . X Old Index No. of the year


In other words, the new series of index numbers is obtained on multiplying the old index numbers with a

common factor:. 100 .


Illustration. From the index number given below, find out index numbers by shifting base from 1960 to 1979.

Year... 1960 1961 1962 . 1979 1980 1981 1982

index number 100 76 68 .. 50 60 70 75

4.6 Splicing of Indexes

Years Index No. Index Number

with base 1960

1960 100100X 100=200

50

1961 7676 X 100=152

50

1962 6868 Xl00=136

50

1979 5050 X 100=100

50

1980 6060 X 100=120

50

1981 7070 X 100=140

50

1982 7575 X 100=150

50


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A series of index numbers may be discontinued because of obsolete commodities included in it or

because of change in weights of these commodities.If a new series of index numbers is constructed with

changed commodities or changed weights, the two series (old and new) are not comparable. The oldand new series must therefore be adjusted so that the two series are comparable.

To adjust the new series, new index numbers are multiplied by the ratio of the old to the new index inthe period of discontinuation:

Likewise, to adjust the old series, old index numbers are multiplied by the ratio of the new to old index:

The above procedures are known as Splicing Index Numbers.

Example:

In the data given below, 2001 is the year of discontinuation of the old series. Construct a continuous

series by splicing:

(a) Old series, and

(b)New series.

Year Index(Old Series)

Index(New Series)

1995 99.8

1996 96.7

1997 95.3

1998 111.91999 134.6

2000 159.8

2001 173.2 96.72002 100.0

2003 100.9

2004 109.12005 111.0

Solution:

(a) Old Series: To splice the old series, multiply old indices by 0.5583, i.e.,

.


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(b) New Series: To splice the new series, multiply new indices by 1.7911, i.e., .

YearIndex

(Old)

Index

(New)

Spliced

Index

(Old)

Spliced

Index

(New)1995 99.8 55.7 99.8

1996 96.7 54.0 96.7

1997 95.3 53.2 95.31998 111.9 62.5 111.9

1999 134.6 75.1 134.6

2000 159.8 89.2 159.82001 173.2 96.7 96.7 173.2

2002 100.0 100.0 179.1

2003 100.9 100.9 180.72004 109.1 109.1 195.4

2005 111.0 111.0 198.8

4.7 Deflating of Index Number

4.8 Uses of Index Numbers

1. Establishes trends : Index numbers when analyzed reveal a general trend of the phenomenon under

study. For eg. Index numbers of unemployment of the country not only reflects the trends in the

phenomenon but are useful in determining factors leading to unemployment.

2. Helps in policy making : It is widely known that the dearness allowances paid to the employees is linkedto the cost of living index, generally the consumer price index. From time to time it is the cost of living

index, which forms the basis of many a wages agreement between the employees union and the

employer. Thus index numbers guide policy making.

3. Determines purchasing power of the rupee : Usually index numbers are used to determine the purchasing

power of the rupee. Suppose the consumers price index for urban non-manual employees increased from

100 in 1984 to 202 in 1992, the real purchasing power of the rupee can be found out as follows:

100/202=0.495

It indicates that if rupee was worth 100 paise in 1984 its purchasing power is 49.5 paise in 1992.

4. Deflates time series data : Index numbers play a vital role in adjusting the original data to reflect reality. For

example, nominal income(income at current prices) can be transformed into real income(reflecting the actual

purchasing power) by using income deflators. Similarly, assume that industrial production is represented in value

terms as a product of volume of production and price. If the subsequent years industrial production were to be

higher by 20% in value, the increase may not be as a result of increase in the volume of production as one would

have it but because of increase in the price. The inflation which has caused the increase in the series can be


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eliminated by the usage of an appropriate price index and thus making the series real.

4.9 Limitations of Index Numbers.

Although index numbers are indispensable tools in economics, business management etc., (they have their

limitations and proper care should be taken in using and interpreting them. Some of their limitations are

enumerated below:

1. Since index numbers are computed from sample.data, all the errors inherent in any sampling procedure creep

in its construction. Hence the index numbers reflect only approximate changes in the relative level of a

phenomenon.

2. .At each stage of the construction of the index numbers, starting from selection of commodities to the choice

of formula there is likelihood of the error being introduced. An attempt should be made to minimise these errors,

as far as, possible.

3. Due to rapid advancements in science and technology these days, there is a rapid change in the tastes,

customs and fashions and consequently in the pattern of consumption of various commodities among the people

in a society. Hence index numbers, may not be able to keep pace with the changes in the nature and quality of

the commodities consumed at the two periods being considered and hence may not be truly representative.

4. None of the formula for the construction of index numbers is exact and contains the so-called 'Formula error'

for example; Laspeyer's index has an upward bias, while Paasche's index has a downward bias.

5. Index numbers are special type of averages. Since the various averages (mean, median, geometric. mean)

have their relative limitations, their indiscriminate use may also introduce some error.

6. By subjective selection of base year, commodities, price 'and 'quantity quotations, index numbers are liable to

be manipulated by selfish persons to obtain the desired results.

In spite of the above limitations, the index numbers if properly constructed with caution are extremely useful

devices.

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9c462Index Number