UNIT 10INDEX NUMBERS

INTRODUCTION:Change is the order of the day. Every economic factor whether it be price, value of money, production,

sales or profits changes from time to time. So, we have to deal with the average of changes in a group of related variables, that may relate to periods of time or between places. Generally, the changes are studied in respect of time rather than place. These changes mainly due to purchasing power of money. That is value of money changes now and them. The changes of money value cannot be measured directly. It depends upon the price level. This change in price level is measured more effectively through index numbers.

BACK GROUNDThe index numbers were first introduced in Italy in the year 1764. The first index was constructed to

compare the Italian price index 1750 with the price level of 1500, it spreaded to other countries later. Now index number techniques are used for the measure of various economic and business activities.

Index numbers today are the most widely used statistical device for estimating trends in prices, wages production and other economic variables. Though originally developed to measure the effect of changes in prices, there is hardly any field, now – a – days where index numbers are not used.

Meaning & Definition.Index numbers is a combination of two words namely index meaning an indicator and number meaning

a numerical figure. In other words index number indicates an increase or decrease of prices, value of money, production, sales etc in a particular period as compared to some previous period.

An index Number is a number which is used to measure the level of a certain phenomenon as compared to the level of the same phenomenon at some standard period.

An index number is a quantity which, by reference to abase period, shows by its variation, the changes in the magnitude over a period of time.

Definitions:-Let us consider the following definitions

A.M. TUTTLE says “ index number is a single ratio which measure the combined change of several variables between two different times, places of situations”.

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

Index number are devices for measuring differences in the magnitude of a group of related variables - Croxton & Cowden. Characteristics

On the basis of the study and analysis of definitions of index numbers, the following points are worth considering.

1. Index numbers are specialized averageIndex numbers help in comparing the changes in variables which are in different units

2. Index number are expressed in percentage Index number are expressed in terms of percentages so as to show the extent of changes

3. Index number measure changes not capable of direct measurement.Where it is difficult to measure the variation in the effects of a groups of variables directly, but

relative variation are measured the help of index numbers 4. Index number is for comparison

The index number by their nature is comparative. They compare changes taking place over time or between places

USER OF IMDEX NUMBERS Index numbers are today one of the most widely used statistical devices. They are particularly useful

measuring relative changes. Some of important user of the index number is show below:-

1. They measure the relative changes:

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Index number are particularly useful in measuring relative change. They give better idea of changes in levels of prices, production, and business activity etc.2. They are of better comparison

The index number reduce the changes of price level into more useful and understand able from. The number of the changes are further reduce to percentage which are easily comparable.3. They are good guides

Index number are not restricted to the price phenomenon alone. Any phenomenon, which is speared over a period to time is capable of being expressed numerically through index number. Thus various kinds of index numbers serve different user4. They are economic barometers

Various index numbers compute d for different purposes are of immense value in dealing with different economic problems. Thus index number are the economic barometers.5. They are the pules of the economy

The stability of price or their inflating or deflating condition can well be observed with the help of indices. Index number of general price level will be measure the purchasing power of money.6. They are the wage adjuster

In all fields of economy the wage adjustment are done with the study of consumer price index numbers. Dearness allowance of the employee of the public and private enterprises are the determined on the basis of consumer price index number. 7. Index number help to compare the stranded living of different classes of people 8. They are a specialized type of averages 9. They help in formulating polices.10. They measure the purchasing power of money.11. Index number serves as a guide to make the relevant decisions for example, investment index

number like NSE, BSE, etc are of great help to those interested in the stock market

General principles, problems in the constructions of index numbers The following general principles are to be carefully adopted in the construction of

Index numbers1. Purpose or object

The statistician must clearly determine the purpose for which the index number are to be constructed because there is no all purpose index numbers. Every index number has got its own uses & liabilities nature of information likely to be collected and the method to be followed depend meanly on the purpose and the scope of the index numbers.2. Selection of base period

A base year is one with reference to which price changes in the current year are expressed. We can not say whether price level current year is increased or decreased unless the price level of the current year is compared with the price level of the base year. There fore the base year must be carefully selected. It must be a normal year and should be free from all kinds of abnormalities like the effect of inflation, deflation was earthquake distant from the current year.

Base year may be fixed base or chain base .in case off fixed base we will take a particular year as the base year but in case of chain base the previous year will Become the base year for the succeeding year.3. Selection of commodities:

The number of commodities that should be included in the index number depends largely on the purpose of the index number. The commodities selected should be representative of the tastes, habits, customs and necessities of the people to whom the index number relates.4. Ascertaining-market prices:

Usually there are two kinds of price prevailing in a market that is whole sale price and the retail price. Whole sale price is more stable than retail price whole sale price is used when index number is constructed to measure the cost of living of all class of people. Where as retail is used when index number is constructed to measure the cost of living of particular class of people.5. Selection of appropriate average:

Index number can be constructed with the help of mean geometric mean. The geometric mean is most appropriate average for the contraction of the index number.

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6. Choosing a formulae for index number.A large number of formula have been devised for the contraction of the index number which work

under different assumption and give different results the choice of suitable formula in a given situation world depend upon the purpose of constructing index number and the data suitable for the same as such no one particular formula can be regarded as best under all circumstances.

CONSTRUCTION OF INDEX NUMBERThe various methods of construction of index number are given below METHODS INDEX NUMBER

UNINEIGHTED IN WEIGHTED IN

SIMPLE SIMPLEAVERAGE WEIGHTED WIEGHTEDAGGREGATE OF PRICE RELATIVE AGGREGATE AVERAGE OF PRICE RELATIVE

Un-weighted index numbersConstruction of index numbers, without assigning corresponding weights this process of constructing

index numbers involves two method.1. simple aggregate method2. simple average of price method

Simple Aggregate method:This is the simplest method of constructing the index numbers. The prices of the different

commodities of the current year are added and the total is divided by the sum of the prices of the base year. Commodity and multiplied by 100.

Symbolically Po1 = P 1 x 100 P0

Pol = price index for the current year with reference to the base year.P1= Aggregate of prices for the current yearP0 = Aggregate of prices for the base year.

ILLUSTRATION =01Construct an index number for 2002 taking 2001 as base.

Commodities Price in Rs2001

Prices In Rs2002

ABCD

90409030

9560

11035

SolutionConstruction of price index

formula Pol= P 1 x 100

P0

Commodities Base year2001

Current year2002

Po P1

A-B-CD

90409030

9560

11035

Po 250 P1 300

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Pol = P 1 x 100 p0

= 300 x 100 250 = 120 %

This means that the prices have increased in the year 2002 to the extent of 20% when compared with the prices of 2001

Defects in simple Aggregate MethodLargely numbered figures largely influence the index number. The relative importance of various

commodities is not taken into account in the index as it is un weighted.

2. SIMPLE AVERAGE OFPRICE RELATIVE METHODIn this method, the price relative of each item is calculated separately and then averaged. A price

relative is the price of the current year expressed as a percentage of the price of the base year, when mean is used.

Symbolically Po1 = P n where P = P1 x 100

P0

n = No/of commodities.

When the geometric mean is used. Po1=Antilog logp

nILLUSTRATION = 02

Compute a price index for the following by a. Simple Aggregateb. Average of price relative method, busing both meant and Geometric Mean.

Commodity A B C D E FPrice in Rs 2001 20 30 10 25 40 50Price in Rs 2003 25 30 15 35 45 55Solution

Calculation for price index

Commodity PricePo

PriceP1

Price relativeP1/poX100=P

Log p

ABCDEF

203010254050

253015354555

25/20X100=12530/30X100=10015/10X100=15035/25X100=140

45/40X100=112.555/50X100=110.0

2.09692.00002.17612.14612.05112.0414

175 205N=6 737.5

p 12.5116

When G.M is usedPo1 = logp n =A.L of 12.5116 6=111.7

1. Simple Aggregate MethodPo1= P 1 x100 Po

= 205 x 100 = 117.14% 175

2.Average Price Relative MethodPo1= p = 7375 =122.91

n 6

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Merits of simple Average of price Relative method1. It gives equal importance to all items2. Extreme items do not unduly affect the index3. The influence due to different units is completely removed

Demerits1. The use of geometric mean involves difficulties of computation.2. It fails to give any consideration to the relative importance of different items.

WEIGHTED INDEX NUMBERS The purpose of weighting is to make the index numbers more representative and to give more important to them: weighted index numbers are of two types.

1. Weighted aggregate index numbers2. Weighted average of price relative

Weighted aggregate index numbersAccording to this method, price themselves are weighted by quantities Pxq. Thus physical quantities are

used as weights. There are various methods of assigning weights and thus various formulas have been formed for the construction of index numbers.

Some of the important formulas are given below1. Laspeyre’s method2. Paasche’s method3. Dorbish and Bowley’s method4. Fisher’s ideal method

LASPEYRES METHOD In this method the base year quantities are taken as weights. This index number an upward bias. That

is when prices increases there is a tendency to reduce the consumption of higher priced goods. This I N is widely used in practical work

Symbolically Po1= P 1qo x 100 Poqo

Steps 1. Multiply the current year prices of various commodities with base year weights and obtains P1qo2. Multiply the base year prices of various commodity with base year weight & obtain Poqo3. Use the formula.

PAASCHE’S PRICE INDEX NUMBER In this method the current year quantities are taken as weight

Symbolically Po1 = P 1q1 x 100 Poq1

Steps 1. Multiply the current year prices of various commodities with current year weights and obtain P1q1 2. Multiply the base year prices of various commodities with the current year weights and obtain P0q1 3. Use the formula

This index number has a downward bias. This formula is not used frequently in practice where the number of commodities is large.

DORBISH AND BOWLEY’S PRICE INDEX NUMBER.Dorbish and Bowley have suggested the arithmetic mean of the Laspeyre’s price index and Paasche’s

price index in order to eliminate the influence of high and low prices. The Dorbish & Bowlers formula for constructing price index number is.

Po1=L+ P x 100 or 2

P 0q1 + P 1q1 P 0q0 P 0q1 x 100 2

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FISHER’S IDEAL INDEX NUMBERProf. IRWING Fisher has suggested a compromise between Laspeyre’s and Paasche’s formula by taking

geometric mean of these formula thus fishes formula for price index is given by.

Po1 = L x P x 100OR

Po1 = P 1q0 x P 1q1 x 100P0q0 P0q1

Merits1.This formula takes in to account both current year as well as base year prices and quantities.2.It is free from bias, upward as well as downward.3.It statistics both the time reversal test as well as the factor several test. That is why it is called an ideal formula.

Demerits1. Thus formula is difficult to interpret.2. It requires the prices and quantities for base year current year.3. It is not, however, a practical index to compare because it is excessively laborious.

ILLUSTRATION = 03From the following data construct price index using

1. Laspeyre’s Method2. Paasche’s Method3. Dorbish & Bowley’s Method.4. Fisher’s ideal index number.

CommoditiesBase Year Current Year

Price in Rs Quantity in Kg Price in Rs. Quantity in KgABCDE

534

117

50100603040

1046

1410

56120602436

Solution Calculation for index numbers

ItemsBase year current year

Po qo P1 q1 P1qo Poq0 P1q1 P0q1

ABCDE

534117

50100603040

10461410

56120602436

500400360420400

250300240330280

560480360336360

280360240264252

2080 P1q0 1400 Poqo 2096 P1q1 1396 Poq1

1. Laspeyre’s Method Po1 = P 1qo x 100 = 2080 x 100 = 148.57

Poqo 14002. Paasche’s Method Po1 = P 1q1 x 100 = 2096 x 100 = 150.14% P0q 0 1396

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3. Dorbish & Bowley’s method Po1 = P 1qo + P 1q1

P 0q0 + P 0q1 x 100 2

= 2080 +2096 1400 1396 x 100 =1.493 x 100 =149.3

2

4. Fisher’s ideal index Number Po1= P 1qo x P 1q1 x 100

P0q0 Poq1 = 2800 x 2096 x 100 = 1.4857 x1.5014 = 149.35%

1400 1396

ILLUSTRATION = 04From the following data, construct price index using

1. Laspeyre’s method2. Paasche’s method3. Fisher’s ideal I N.

Commodities 2001 2002Price in Rs Value in Rs Price In Rs Value in Rs

A 5 50 6 72B 7 84 10 80C 10 80 12 96D 4 20 5 30E 8 56 8 64

SolutionSince we are given price and value, first divide values price and obtain quantity figures and then apply

formulae.Quantity = value/price

Commodities2001 2002

Price Quantity Price QuantityP0 q0 P1 q1 P1qo P0q0 P1q1 Poq1

ABCDE

57

1048

1012857

6101258

128868

60120962556

5084802056

7280963064

6056802464

357 P1qo

290 P0q0

342 P1q1

284 Poq1

1. Laspeyre’s Method Po1 = P 1qo x 100 = 357 x 100 = 123.10%

Poqo 2902. Paasche’s Method Po1 = P 1q1 x 100 = 342 x 100 = 120.4% P0q 0 284 4. Fisher’s ideal index Number

Po1= P 1qo x P 1q1 x 100P0q0 Poq1

= 357 x 342 x 100 = 1.231 x 1.204 =121.74% 290 284

170

ILLUSTRATION = 05Construct price index number from the following data by apply.

1. Laspeyre’s method 2. Paasche’s method 3. Fisher’s ideal method

Commodities A B C D E F

2002 PriceQuantity

2 5 4 2 6 720 40 80 60 100 50

2003 PriceQuantity

3 6 5 3 8 1030 50 90 70 120 70

Solution; - Calculation for the index numbers

Commodities2002 2003

P1q0 P0q0 P1q1 Poq1

Price Quantity Price QuantityPo qo P1 q1

ABCDEF

254267

20408060

10050

3653810

305090701270

60240400180800500

40200320120600350

90300450210960700

60250360140720490

2180P1qo

1630Poqo

2710 P1q1

2020Poq1

1. Laspeyre’s Method Po1 = P 1qo x 100 = 2180 x100 = 133.74%

Poqo 16302. Paasche’s Method Po1 = P 1q1 x 100 = 2710 x 100 = 134.2% P0q 0 2020 4. Fisher’s ideal index Number

Po1= P 1qo x P 1q1 x 100P0q0 Poq1

= 2180 x 2710 x 100 = 1.337 x 1.342 =133.94% 1630 2020Test of consistency of index number

Several formula have been used for the construction of index number. The question arises as to which formula is appropriate to a given problem. A number of tests have been developed and the important among them are.

1.Time reversal test 2.Factor reversal test

Time reversal testReversibility is an important property than an index number should posses. A good index number

should satisfy the time reversal tests. In the words of lrwing Fisher “The formula for calculating an index number should be such that it gives the same ratio between one of point comparison and the there no matter which of two is taken as the base, putting it in another way, in the index number reckoned forward should be reciprocal of one reckoned back ward. One of the advantages claimed in favour of Fisher’s formula is that is make the index number reversible. The time reversal test shows that the following equation hold good symbolically

Po1 x P1o=1 (Excluding the factor 100from each formula)Fisher’s ideal formula satisfies the Time reversal testPo1= P 1qo x P 1q1 x 100

P0q0 Poq1

Reverse time aspect , Keeping factor as constant P10= P oq1 x P 0q0

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P1q1 P1q0

Po1 x P10= P 1qo x P 1q1 x P oq1 x P 0q0 =1 = 1 P0q0 Poq1 P1q1 P1q0

It show that the time reversal test is satisfied

FACTOR REVERSAL TEST Another basic test is that the formula for index number ought to permit interchanging the price and

quantities without giving inconsistent result i.e. the two result multiplied together should give the true value ratio. A good index number should satisfy not only the time reversal test but also the factor reversal test .A good index number should allow time reversibility, interchange of the base year and the current year with out giving inconsistent result.Po1 x q01 = P 1q1

Poqo

Po1= P 1qo x P 1q1 x 100P0q0 Poq1

Reverse factor, keeping time aspect as constant.qo1= q 1P0 x q 1P1

q0P1 q0P1

Po1 x q01= P 1qo x P 1q1 x q 1q0 x q 1P0 P0q0 Poq1 q0P0 q0P1

It shows ==(P1q1/ P0q0)2 = P1q1/P0q0

ILLUSTRATION =06Compute index number using Fisher’s ideal formula and show that it satisfies time reversal test and

factor reversal test.

Commodity2003 2004

price quantity price quantity

ABCDE

624

108

50100603040

1026

1212

56120602436

Solution Construction of fisher ideal index number

CommoditiesBase year Current year

P1qo Poqo P1q1 Poq1Price Quantity Price QuantityP0 p0 P1 q1

A 6 50 10 56 500 300 560 336B 2 100 2 120 200 200 240 240C 4 60 6 60 360 240 360 240D 10 30 12 24 360 300 288 240E 8 40 12 36 480 320 432 288

1900P1qo

1360Poqo

1880P1q1

1344poq1

Fisher’s ideal indexPo1= P 1qo x P 1q1 x 100

P0q0 Poq1

1900 x 1880 x 100 = 1.397 x 1.3988 = 139.789

172

1360 1344Application of time reversal test TRT

P01 x P10 =1Po1 x P10= P 1qo x P 1q1 x P oq1 x P 0q0

P0q0 Poq1 P1q1 P1q0

Po1 x q01= 1900 x 1880 x 1344 x 1360 = 1 = 1 1360 1344 1880 1900

P01 x P10 = 1 TRT is satisfied by fishers index formula.

Application of FRTPo1 x q01= P 1qo x P 1q1 x P 1q0 x q 1P0

P0q0 Poq1 q0P0 q0P1

= 1900 x 1880 x 1344 x 1880 1360 1344 1360 1900 = (1880/1360)2 =1880/1360

Fishers index satisfies factor reversal Test also.

ILLUSTRATION = 07Calculate Fishers ideal index from the following data and prove how it satisfies the T R & FRT

Commodity e 2003 2004Price in Rs Expenditure in Rs Price in Rs Expenduture in Rs

A 16 160 20 240B 20 240 24 192C 10 80 10 100D 8 112 6 138E 40 200 50 300

SolutionSince we are given price and value first divide values by price and obtain quantity figure and then apply

formula.Quantity = Expenditure / price

Calculation of index Numbers

Commodities

2003 2004

P1qo Poqo P1q1 Poq1

Price Quantity Price QuantityPo qo P1 Q1

A 16 10 20 12 200 160 240 192B 20 12 24 8 288 240 192 160C 10 8 10 10 80 80 100 100D 8 14 6 23 84 112 138 184E 40 5 50 6 250 200 300 240

902P1qo

792Poqo

970P1q1

876Poq1

Calculation of fishers index NumberPo1= P 1qo x P 1q1 x 100

P0q0 Poq1

902 x 970 x 100 = 1.138 x 1.107 = 1.122 x 100 = 112.2 792 876

Application of time reversal test TRTP01 x P10 =1

Po1 x P10= P 1qo x P 1q1 x P oq1 x P 0q0 P0q0 Poq1 P1q1 P1q0

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Po1 x q01= 902 x 970 x 876 x 792 = 1 = 1 792 876 970 902It satisfies the conditions of TRT

Application of FRTP01 x P10 = P1q1/P0q0

Po1 x q01= P 1qo x P 1q1 x P 1q0 x q 1P0 P0q0 Poq1 q0P0 q0P1

Substitute their values = 902 x 970 x 876 x 970

792 876 793 902 = (970/792)2 = 970/792

Its Satisfies the conditions of FRT

ILLUSTRATION = 08From the following data, calculate the price index number using

a. Laspeyre’s Methodb. Fisher’s ideal index numberc. Also show how it satisfies time and Factor Reversed Test.

Items A B C D E F

Year Price Rs

ValueRs

PriceRs

ValueRs

PriceRs

ValueRs

PriceRs

ValueRs

PriceRs

ValueRs

PriceRs

ValueRs

2002 15 1200 16 1600 15 900 18 720 14 840 12 6002003 18 1620 20 2800 22 1540 20 1000 15 1200 15 1350

SOLUTIONSince we are given price and value first divide values price and obtain quantity figures and then apply

formula

Commodities

2002 2003Price

Po

Quantity qo

PriceP1

Quantity q1

P1qo Poqo P1q1 Poq1

A 15 80 18 90 1440 1200 1620 1350B 16 100 20 140 2000 1600 2800 2240C 15 60 22 70 1320 900 1540 1050D 18 40 20 50 800 720 1000 900E 14 60 15 80 900 840 1200 1120F 12 50 15 90 750 600 1350 1080

7210P1qo

5860Poqo

9510P1q1

7740Poq1

Laspeyre’s Method Fisher’s index numberPo1 = P 1qo x 100 = 7210 x 100 = 123.03% Poqo 5860

Po1= P 1qo x P 1q1 x 100P0q0 Poq1

7210 x 9510 x 100 5860 7740

= 1.230 x1.228 = 122.9%

Application of time reversal test TRTP01 x P10 =1

Po1 x P10= P 1qo x P 1q1 x P oq1 x P 0q0 P0q0 Poq1 P1q1 P1q0

Po1 x q01= 7210 x 9510 x 7740 x 5860 = 1 = 1

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5860 7740 9510 7210It satisfies the conditions of TRT

Application of FRTPo1 x q01= P 1qo x P 1q1 x P 1q0 x q 1P0

P0q0 Poq1 q0P0 q0P1

Substitute their values = 7210 x 9510 x 7740 x 9510

5860 7740 5860 7210 = (9510/5860)2 = 9510/5860

It satisfies both TRT & FRT

COST OF LIVING OR CONSUMER PRICE INDEXConsumer price index numbers are designed to measure the average change over time in the price paid

by ultimate consumer for a specifies quantity of good and services. Consumer price indices measure the change in the cost of living of workers due to change in the retail price. A change in the price level affects the cost of living of different classes of people differently. The general index number fails to reveal this. So there is the need to construct consumer price index people consume different types of commodities. People’s consumption rabit is also different from man to man, place to place etc.

The scope of consumer price is necessary to specify the population groups covered for ex working class, middle class, poor class or rich class etc.

USES OF CONSUMER PRICE INDEX

1. This is very useful in wage negotiations and wage contracts and allowances adjusted in many countries.2. Govt. can make use of these indices for wage policy, price policy, Taxation, general economic policies

etc.3. Changes in the purchasing power of money and real income can be measured.4. We can analyse the market price for particular kinds of good and services by this index.

Methods of constructing consumer price index.There are two methods of constructing consumer price index.1. Aggregate expenditure method2. Family Budget method, or Method of weighted Relation.

1. Aggregate expenditure methodThis method is based upon the Laspeyre’s method. It is widely used, the quantities of commodities

consumed by a particular group in the base year are the weight.The formula is – consumer price index = P 1qo x 100

Poq0

2. Family Budget methodHere an aggregate expenditure of an average family an various items is estimated and it is value weight.

The formula is Consumer price index = PV

VWhere P = P1 x 100 for each item

Po

V = Value weight i.e Poqo

Weighted average price relative method and Family budget method are the same for finding out consumer price index.

ILLUSTRATION =- 09Construct the consumer price index number for 2004. On the basis of 2003 from the following data

using.1. Aggregate Expenditure method.2. Family budget method

Quantity consumedPrice in Rs Price in Rs

175

A 6 quintals 5.75 6.00B 6 quintals 5.00 8.00C 1 quintals 6.00 9.00D 6 quintals 8.00 10.00E 4 kg 2.00 1.50F 1 quintals 20.00 15.00

SOLUTION FOR AEM FBM

commodities qo P0 P1 P1qo Poqo P1 x 100 = PPo

YP0q0

PV

A 6 5.75 6.00 36 34.50 (6/5.75) x 100=104.34 34.5 3599.99B 6 5.00 8.00 48 30.00 (8/5) x 100 =160.00 30.0 4800.00C 1 6.00 9.00 9 6.00 (9/6) x 100 =150.00 6.0 900.00D 6 8.00 10.00 60 48.00 (10/8) x 100 = 125.00 48.0 6000.00E 4 2.00 1.15 6 8.00 (1.5/2) x 100 = 75 8.0 600.00F 1 20.00 15.00 15 20.00 (15/20) x100 = 75 20.0 1500.00

174P1qo

146.50poqo

146.5v

17399.99PV

I. AEMI C.P.I = P1qo x 100 = 174 x 100 = 118.77

P0q0 146.5

II FBMC P I = PV = 17399.99 = 118.77

V 146.5

ILLUSTRATION = 1 0Calculate Consumer price index from the following data. Using Family Budget Method.

Articles Quantity in 2000 Unit Price in Rs 2000 Price in Rs 2001Rice 2 quintals Per quintal 500 750

Wheat 75 Kg P/quintal 400 560Oil 25 liters 10 liters 2.80 350

Sugar 40 Kgs Kg 2.50 3.50Pulses 30 Kgs Kg 12 15fuel 2 Tonnes Ton 50 60

Misce 25 u nits Unit 4.40 5.50

SOLUTION

ArticlesQuantity

Unit

Prices in Rs

Price in Rs 2001

(P1/Po) x100 =P V

P0q0PVIn 2000

q0

2000Po

P P

Rice 2Q P/Q 500 750 150 1000 150000Wheat 75Kg P/Kg 4 5.60 140 300 42000

Oil 25Lars P/Lt 28 35 125 700 87500Sugar 40Kgs P/Kg 2.50 3.50 140 100 14000

176

Pulses 30Kgs P/Kg 12 15.0 125 360 45000Fuel 2 tonnes P/T 50 60 120 100 12000

Misce 25units unit 4.40 5.50 125 110 13750

2670V 364250PV

Note 1quantal =100kg1990- wheat per kg =400/100 = Rs 41996 Per kg =560/100 = 5.601990 Per litre =280/10 =281996 Per litre =350/10 =35 C P.1 = PV =364250 =136.42

V 2670

ILLUSTRATION = 11 Construct the cost of living index number for 2003 on the basis of 2002 from the following data

a . Aggregate expenditure method b. Family budget method

commodities Quantity used in 2002 Unit Price in Rs 2002 Price in Rs 2003

Rice 200kg Per quintal 1500 1800Wheat 150 kg P/quintal 110 1200Jowar 100kg P/quintal 900 1000Pulser 50 kg P/quintal 2500 3000Suger 20 kg P/quintal 1300 1400

Oil 20 liter Per40liter 1000 1200Fire wood 40 mounds Per mounds 30 50

Dal 10 kg Per kg 20 25Kerosin oil 10 liter Per 20liter 60 80

Elolt 50 meter Per meter 10 15SOLUTION Calculation for index number

Commodities

Quantity used

In 2002qoUnit 2002

Po2003P1 P1qo Poqo (P1/Po)

x 100=P V PV

Rice 200kg P/kg 15 18 3600 3000 120 3000 360000Wheat 150kg P/kg 11 12 1800 1650 109 1650 180000Jower 100kg P/kg 9 10 1000 900 111 900 100000Pulses 50kg P/kg 25 30 1500 1250 120 1250 150000Sugar 20kg P/lt 13 14 280 260 107.6 260 28000

Oil 20lit P/d 25 30 600 500 120 500 60000Firewood 40md P/mt 30 50 2000 1200 166.7 1200 20000

Dal 10kg P/kg 20 25 250 200 125 200 25000Kerosine oil 10lit P/lt 3 4 40 30 133 30 4000

Cloth 50mts P/mt 10 15 750 500 150 500 7500011820P1qo

9490Poqo

9490V

1182000PV

I. Aggregate expenditure MethodC.L.I = P1qo x 100 = 11820 x 100 = 124.55

P0q0 9490II FBMC L I = PV = 1182000 = 124.55

V 9490

ILLUSTRATION =12An enquiry into the budgets of middle class families in Bungler gave the following information.

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What changes in the cost of living index of 2001 have taken place as compared to 1999.Expenses Food Rent Clothing Fuel MiscellaneousPercentage 1991 35 15 20 10 20Price relative 2001 116 120 125 125 150SOLUTION

Items of expenditure Weights V P PVFood 35 116 4060Rent 15 120 1800Clothing 20 125 2500Fuel 10 125 1250Miscellany 20 150 3000

100 636 126.10Note:- Given percentages are taken as weights.

Price Relatives = Index – P CLI = PV = 12610 = 126.1%

V 100

THEORY QUESTIONS (5, 10 & 15 Marks)1. What is an Index Number? State the uses of index numbers.2. Why are index numbers called “Economic Barometers”?3. What is consumer price index number? Briefly explain methods used for construction of CPI4. What do you mean by TRT and FRT? Show how Fisher’s formula satisfies these tests5. What are the uses of consumer price index number?6. Briefly explain the problems in the construction.7. Explain briefly the important methods of construction of index number. State the utilities of index

number.8. What are the characteristics of index numbers.

Practical ProblemsProblem No 1

Given the following data, calculate data, calculate price index Number through Fisher’s ideal I n dex Method and test the consistency of it by ( a ) TRT and ( b ) FRT.

Commodities 1999 2001Price Rs Value Rs Price Rs Value Rs

A 30 600 40 1000B 60 1500 120 2400C 25 450 40 800D 15 150 25 200E 30 900 50 1600

Note = quantity = price, [ Answer FIN = 172.07]

Problem No:2Construct the cost of living index Number using.

1. Aggregate Expenditure Method 2. Family Budget Method.Commodities Quantity in 2001 2001 price in Rs 2002 Price in Rs

A 100 8-00 12B 25 6-00 7.50C 10 5-00 5-25D 20 48-00 52.00E 65 15-00 16-50

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F 30 9-00 27-00Answers: A E M = CPI = 124.5

F B M = CPI = 124.5Problem No=03

An enquiry into the budgets of middle class families in a certain city gave the following information calculate cost of living index.

Expenses on Food Fuel Clothing Rent MisecllParticular % age 35% 10% 20% 15% 20%

Price 2001 150 25 75 30 40Price 2002 145 23 65 30 45

[Answers CLI= 97.57]

Problem No = 4Calculate Fisher’s index from the following data and show that it satisfies both TRT And FRT.

Commodities A B C D EBase year – price in Rs 10 8 20 18 35Base year – value in Rs 200 108 160 144 280Current year – value in Rs 300 220 250 140 300Current year - quantity 25 22 10 7 10

Note:- In the above problem, base year quantity and current year price are not given. They are obtained with the help of values. [Ans: P01 = 109.5523]

Problem No = 5The following are the group index numbers and the group weights of an average working class family’s

budget. Construct the cost of living index number.Group Index number In eights

Food 352 48Fuel and lighting 220 10Clothing 230 8Rent 160 12Misee% 190 15Answer C L I = 276 .41C L I = 25706/93 = 276.41

Problem =6Calculate the cost of living index number from the following table for the year 2004 with 2003 as the

base year.Commodities Unit Quantity 2003 Price in Rs 2003 Price in Rs 2004Rice Per Kg 20 Kg 1-00 2-00Wheat Per Kg 50 Kg 0.60 1-10Oil Per Kg 10 Kg 2.00 4-00Ghee Per Kg 500 Kg 8.00 14-00Sugar Per Kg 5 K g 1.00 1-80Cloth Per miter 40 Miter 2.00 3-75House rent - One house 4.00 75-00

Answer :- C L I = 376/199 X 100 = 188.94

Problem = 07From the following data ,construct price index, using

a. Laspeyre’s Methodb. Paasche’s Methodc. Fishers Ideal index number

Commodities Base year Current year Base year Current yearPrice Price in Rs Quantity Quantity

L 12 20 100 120M 4 4 200 240

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N 8 12 120 150O 20 24 60 50[Answers:- LIN = 136.5, PIN = 138.26 FIN = 137.4]

Problem No:- 8Calculate Fishers ideal Number from the following data and show that the time and Factor Reversal

Tests are satisfied by this index number.Commodities Base year Current year

A 6 300 10 560B 2 200 6 240C 4 240 6 360D 10 300 12 288E 8 320 9 432

[Answer FIN = 152.9%]

Problem No=9 From the following data calculate the price index number usinga. Laspeyre’s methodb. Fisher’s ideal index Numberc. Also show how it satisfies time & Factor Reversal Test.

Commodities A B C D E

Year Price Rs

ValueRs

Price Rs

ValueRs

Price Rs

ValueRs

Price Rs

ValueRs

Price Rs

ValueRs

2001 16 1280 17 1700 16 960 19 760 15 9002002 19 1710 22 3080 23 1610 21 1050 16 1280

[Answer LIN=123.2 FIN = 123.1]

Problem No:10From the following data, construct cost of living index, by using

1. Aggregate Expenditure Method.2. Family Budget Method.

Commodities Quantity used inBase year

Base yearPrice Rs

Current yearPrice In Rs

Wheat 4 8 14Rice 2 15 21Dal 1 10 14Oil 5 20 30

Ghee 3 6 12Cereals 1 7 14

Vegetables 2 5 15[Answers: L I N = 165.21, F B M I N = 165.21]

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