1.
UNIT 10
INDEX 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 GROUND
The 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 average
Index 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:
164
2.
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 user
4. 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 numbers
1. 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.
165
3.
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 NUMBER
The various methods of construction of index number are given below
METHODS
INDEX NUMBER
UNINEIGHTED IN WEIGHTED IN
SIMPLE SIMPLEAVERAGE WEIGHTED WIEGHTED
AGGREGATE OF PRICE RELATIVE AGGREGATE AVERAGE OF PRICE
RELATIVE
Un-weighted index numbers
Construction of index numbers, without assigning corresponding weights this process of constructing
index numbers involves two method.
1. simple aggregate method
2. 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 = ∑P1 x 100
∑P0
Pol = price index for the current year with reference to the base year.
∑P1= Aggregate of prices for the current year
∑P0 = Aggregate of prices for the base year.
ILLUSTRATION =01
Construct an index number for 2002 taking 2001 as base.
Price in Rs Prices In Rs
Commodities
2001 2002
A 90 95
B 40 60
C 90 110
D 30 35
Solution
Construction of price index
formula
Pol= ∑P1 x 100
∑P0
Base year Current year
Commodities
2001 2002
Po P1
A- 90 95
B- 40 60
C 90 110
D 30 35
∑Po 250 ∑P1 300
166
4.
Pol =∑P1 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 Method
Largely 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 METHOD
In 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
n
ILLUSTRATION = 02
Compute a price index for the following by
a. Simple Aggregate
b. Average of price relative method, busing both meant and Geometric Mean.
Commodity A B C D E F
Price in Rs 2001 20 30 10 25 40 50
Price in Rs 2003 25 30 15 35 45 55
Solution
Calculation for price index
Price Price Price relative Log p
Commodity
Po P1 P1/poX100=P
A 20 25 25/20X100=125 2.0969
B 30 30 30/30X100=100 2.0000
C 10 15 15/10X100=150 2.1761
D 25 35 35/25X100=140 2.1461
E 40 45 45/40X100=112.5 2.0511
F 50 55 55/50X100=110.0 2.0414
N=6 737.5
175 205 12.5116
∑p
When G.M is used
Po1 =∑logp
n
=A.L of 12.5116
6
=111.7
1. Simple Aggregate Method 2.Average Price Relative Method
Po1=∑P1 x100 Po1=∑p = 7375 =122.91
∑Po n 6
= 205 x 100 = 117.14%
175
167
5.
Merits of simple Average of price Relative method
1. It gives equal importance to all items
2. Extreme items do not unduly affect the index
3. The influence due to different units is completely removed
Demerits
1. 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 numbers
2. Weighted average of price relative
Weighted aggregate index numbers
According 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 below
1. Laspeyre’s method
2. Paasche’s method
3. Dorbish and Bowley’s method
4. 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= ∑P1qo x 100
∑Poqo
Steps
1. Multiply the current year prices of various commodities with base year weights and obtains P1qo
2. Multiply the base year prices of various commodity with base year weight & obtain Poqo
3. Use the formula.
PAASCHE’S PRICE INDEX NUMBER
In this method the current year quantities are taken as weight
Symbolically Po1 = ∑P1q1 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 ∑P0q1 + ∑P1q1
2 ∑P0q0 ∑P0q1 x 100
2
168
6.
FISHER’S IDEAL INDEX NUMBER
Prof. 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 100
OR
Po1 = √∑P1q0 x ∑P1q1 x 100
∑P0q0 ∑P0q1
Merits
1.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.
Demerits
1. 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 = 03
From the following data construct price index using
1. Laspeyre’s Method
2. Paasche’s Method
3. Dorbish & Bowley’s Method.
4. Fisher’s ideal index number.
Base Year Current Year
Commodities
Price in Rs Quantity in Kg Price in Rs. Quantity in Kg
A 5 50 10 56
B 3 100 4 120
C 4 60 6 60
D 11 30 14 24
E 7 40 10 36
Solution
Calculation for index numbers
Base year current year
Items
Po qo P1 q1 P1qo Poq0 P1q1 P0q1
A 5 50 10 56 500 250 560 280
B 3 100 4 120 400 300 480 360
C 4 60 6 60 360 240 360 240
D 11 30 14 24 420 330 336 264
E 7 40 10 36 400 280 360 252
2080 ∑P1q0 1400 ∑Poqo 2096 ∑P1q1 1396 ∑Poq1
1. Laspeyre’s Method
Po1 = ∑P1qo x 100 = 2080 x 100 = 148.57
∑Poqo 1400
2. Paasche’s Method
Po1 =∑P1q1 x 100 = 2096 x 100 = 150.14%
∑P0q 0 1396
169
7.
3. Dorbish & Bowley’s method
Po1 = ∑P1qo + ∑P1q1
∑P0q0 + ∑P0q1 x 100
2
= 2080 +2096
1400 1396 x 100 =1.493 x 100 =149.3
2
4. Fisher’s ideal index Number
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
= √2800 x 2096 x 100 = √1.4857 x1.5014 = 149.35%
1400 1396
ILLUSTRATION = 04
From the following data, construct price index using
1. Laspeyre’s method
2. Paasche’s method
3. Fisher’s ideal I N.
2001 2002
Commodities
Price in Rs Value in Rs Price In Rs Value in Rs
A 5 50 6 72
B 7 84 10 80
C 10 80 12 96
D 4 20 5 30
E 8 56 8 64
Solution
Since we are given price and value, first divide values price and obtain quantity figures and then apply
formulae.
Quantity = value/price
2001 2002
Commodities Price Quantity Price Quantity
P0 q0 P1 q1 P1qo P0q0 P1q1 Poq1
A 5 10 6 12 60 50 72 60
B 7 12 10 8 120 84 80 56
C 10 8 12 8 96 80 96 80
D 4 5 5 6 25 20 30 24
E 8 7 8 8 56 56 64 64
357 290 342 284
∑ P1qo ∑ P0q0 ∑ P1q1 ∑ Poq1
1. Laspeyre’s Method
Po1 = ∑P1qo x 100 = 357 x 100 = 123.10%
∑Poqo 290
2. Paasche’s Method
Po1 =∑P1q1 x 100 = 342 x 100 = 120.4%
∑P0q 0 284
4. Fisher’s ideal index Number
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
= √357 x 342 x 100 = √1.231 x 1.204 =121.74%
290 284
170
8.
ILLUSTRATION = 05
Construct 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
Price 2 5 4 2 6 7
2002
Quantity 20 40 80 60 100 50
Price 3 6 5 3 8 10
2003
Quantity 30 50 90 70 120 70
Solution; - Calculation for the index numbers
2002 2003
Commodities Price Quantity Price Quantity
Po qo P1 q1 P1q0 P0q0 P1q1 Poq1
A 2 20 3 30 60 40 90 60
B 5 40 6 50 240 200 300 250
C 4 80 5 90 400 320 450 360
D 2 60 3 70 180 120 210 140
E 6 100 8 12 800 600 960 720
F 7 50 10 70 500 350 700 490
2180 1630 2710 2020
∑P1qo ∑Poqo ∑ P1q1 ∑Poq1
1. Laspeyre’s Method
Po1 = ∑P1qo x 100 = 2180 x100 = 133.74%
∑Poqo 1630
2. Paasche’s Method
Po1 =∑P1q1 x 100 = 2710 x 100 = 134.2%
∑P0q 0 2020
4. Fisher’s ideal index Number
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
= √2180 x 2710 x 100 = √1.337 x 1.342 =133.94%
1630 2020
Test 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 test
Reversibility 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 test
Po1= √∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
171
9.
Reverse time aspect , Keeping factor as constant
P10= √ ∑Poq1 x ∑P0q0
∑P1q1 ∑P1q0
Po1 x P10= √ ∑P1qo x ∑P1q1 x ∑Poq1 x ∑P0q0 =√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 = ∑P1q1
∑Poqo
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
Reverse factor, keeping time aspect as constant.
qo1= √∑q1P0 x ∑q1P1
∑q0P1 ∑q0P1
Po1 x q01= √ ∑P1qo x ∑P1q1 x ∑q1q0 x ∑q1P0
∑P0q0 ∑Poq1 ∑q0P0 ∑q0P1
It shows ==√(∑P1q1/ ∑P0q0)2 = ∑P1q1/∑P0q0
ILLUSTRATION =06
Compute index number using Fisher’s ideal formula and show that it satisfies time reversal test and
factor reversal test.
2003 2004
Commodity quantit
price quantity price
y
A 6 50 10 56
B 2 100 2 120
C 4 60 6 60
D 10 30 12 24
E 8 40 12 36
Solution
Construction of fisher ideal index number
Base year Current year
Commodities Price Quantity Price Quantity P1qo Poqo P1q1 Poq1
P0 p0 P1 q1
A 6 50 10 56 500 300 560 336
B 2 100 2 120 200 200 240 240
C 4 60 6 60 360 240 360 240
D 10 30 12 24 360 300 288 240
E 8 40 12 36 480 320 432 288
1900 1360 1880 1344
∑P1qo ∑Poqo ∑P1q1 ∑poq1
172
10.
Fisher’s ideal index
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
√ 1900 x 1880 x 100 = √1.397 x 1.3988 = 139.789
1360 1344
Application of time reversal test TRT
P01 x P10 =1
Po1 x P10= √ ∑P1qo x ∑P1q1 x ∑Poq1 x ∑P0q0
∑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 FRT
Po1 x q01= √ ∑P1qo x ∑P1q1 x ∑P1q0 x ∑q1P0
∑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 = 07
Calculate Fishers ideal index from the following data and prove how it satisfies the T R & FRT
2003 2004
Commodity e
Price in Rs Expenditure in Rs Price in Rs Expenduture in Rs
A 16 160 20 240
B 20 240 24 192
C 10 80 10 100
D 8 112 6 138
E 40 200 50 300
Solution
Since 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
2003 2004
Price Quantity Price Quantity
Commodities Po qo P1 Q1 P1qo Poqo P1q1 Poq1
A 16 10 20 12 200 160 240 192
B 20 12 24 8 288 240 192 160
C 10 8 10 10 80 80 100 100
D 8 14 6 23 84 112 138 184
E 40 5 50 6 250 200 300 240
902 792 970 876
∑P1qo ∑Poqo ∑P1q1 ∑Poq1
Calculation of fishers index Number
Po1= √ ∑P1qo x ∑P1q1 x 100
∑P0q0 ∑Poq1
√ 902 x 970 x 100 = √1.138 x 1.107 = 1.122 x 100 = 112.2
792 876
173
11.
Application of time reversal test TRT
P01 x P10 =1
Po1 x P10= √ ∑P1qo x ∑P1q1 x ∑Poq1 x ∑P0q0
∑P0q0 ∑Poq1 ∑P1q1 ∑P1q0
Po1 x q01= √ 902 x 970 x 876 x 792 = √1 = 1
792 876 970 902
It satisfies the conditions of TRT
Application of FRT
P01 x P10 = ∑P1q1/∑P0q0
Po1 x q01= √ ∑P1qo x ∑P1q1 x ∑P1q0 x ∑q1P0
∑P0q0 ∑Poq1 ∑q0P0 ∑q0P1
Substitute their values
= √ 902 x 970 x 876 x 970
792 876 793 902
= √(970/792) = 970/792
2
Its Satisfies the conditions of FRT
ILLUSTRATION = 08
From the following data, calculate the price index number using
a. Laspeyre’s Method
b. Fisher’s ideal index number
c. Also show how it satisfies time and Factor Reversed Test.
Items A B C D E F
Price Value Price Value Price Value Price Value Price Value Price Value
Year
Rs Rs Rs Rs Rs Rs Rs Rs Rs Rs Rs Rs
2002 15 1200 16 1600 15 900 18 720 14 840 12 600
2003 18 1620 20 2800 22 1540 20 1000 15 1200 15 1350
SOLUTION
Since we are given price and value first divide values price and obtain quantity figures and then apply
formula
2002 2003
Commodi Price Quantity Price Quantity
P1qo Poqo P1q1 Poq1
ties Po qo P1 q1
A 15 80 18 90 1440 1200 1620 1350
B 16 100 20 140 2000 1600 2800 2240
C 15 60 22 70 1320 900 1540 1050
D 18 40 20 50 800 720 1000 900
E 14 60 15 80 900 840 1200 1120
F 12 50 15 90 750 600 1350 1080
7210 5860 9510 7740
∑P1qo ∑Poqo ∑P1q1 ∑Poq1
Laspeyre’s Method Fisher’s index number
Po1 = ∑P1qo x 100 = 7210 x 100 = 123.03% Po1= √ ∑P1qo x ∑P1q1 x 100
∑Poqo 5860 ∑P0q0 ∑Poq1
√ 7210 x 9510 x 100
5860 7740
= √1.230 x1.228 = 122.9%
174
12.
Application of time reversal test TRT
P01 x P10 =1
Po1 x P10= √ ∑P1qo x ∑P1q1 x ∑Poq1 x ∑P0q0
∑P0q0 ∑Poq1 ∑P1q1 ∑P1q0
Po1 x q01= √ 7210 x 9510 x 7740 x 5860 = √1 = 1
5860 7740 9510 7210
It satisfies the conditions of TRT
Application of FRT
Po1 x q01= √ ∑P1qo x ∑P1q1 x ∑P1q0 x ∑q1P0
∑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 INDEX
Consumer 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 method
2. Family Budget method, or Method of weighted Relation.
1. Aggregate expenditure method
This 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 = ∑P1qo x 100
∑Poq0
2. Family Budget method
Here an aggregate expenditure of an average family an various items is estimated and it is value weight.
The formula is
Consumer price index = ∑PV
∑V
Where 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.
175
13.
ILLUSTRATION =- 09
Construct 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 consumed Price in Rs Price in Rs
Commodities
2003 2003 2004
A 6 quintals 5.75 6.00
B 6 quintals 5.00 8.00
C 1 quintals 6.00 9.00
D 6 quintals 8.00 10.00
E 4 kg 2.00 1.50
F 1 quintals 20.00 15.00
SOLUTION FOR AEM FBM
P1 x 100 = P Y
commodities qo P0 P1 P1qo Poqo PV
Po P0q0
A 6 5.75 6.00 36 34.50 (6/5.75) x 100=104.34 34.5 3599.99
B 6 5.00 8.00 48 30.00 (8/5) x 100 =160.00 30.0 4800.00
C 1 6.00 9.00 9 6.00 (9/6) x 100 =150.00 6.0 900.00
D 6 8.00 10.00 60 48.00 (10/8) x 100 = 125.00 48.0 6000.00
E 4 2.00 1.15 6 8.00 (1.5/2) x 100 = 75 8.0 600.00
F 1 20.00 15.00 15 20.00 (15/20) x100 = 75 20.0 1500.00
174 146.50 146.5 17399.99
∑P1qo ∑poqo ∑v ∑PV
I. AEM
I C.P.I = ∑P1qo x 100 = 174 x 100 = 118.77
∑P0q0 146.5
II FBM
C P I = ∑PV = 17399.99 = 118.77
∑V 146.5
ILLUSTRATION = 1 0
Calculate Consumer price index from the following data. Using Family Budget Method.
Articles Quantity in 2000 Unit Price in Rs 2000 Price in Rs 2001
Rice 2 quintals Per quintal 500 750
Wheat 75 Kg P/quintal 400 560
Oil 25 liters 10 liters 2.80 350
Sugar 40 Kgs Kg 2.50 3.50
Pulses 30 Kgs Kg 12 15
fuel 2 Tonnes Ton 50 60
Misce 25 u nits Unit 4.40 5.50
176
14.
SOLUTION
Quantity Prices in Price in (P1/Po)
Rs Rs 2001 x100 =P V
Articles In 2000 Unit PV
2000 P0q0
q0 P P
Po
Rice 2Q P/Q 500 750 150 1000 150000
Wheat 75Kg P/Kg 4 5.60 140 300 42000
Oil 25Lars P/Lt 28 35 125 700 87500
Sugar 40Kgs P/Kg 2.50 3.50 140 100 14000
Pulses 30Kgs P/Kg 12 15.0 125 360 45000
Fuel 2 tonnes P/T 50 60 120 100 12000
Misce 25units unit 4.40 5.50 125 110 13750
364250
2670∑V
∑PV
Note 1quantal =100kg
1990- wheat per kg =400/100 = Rs 4
1996 Per kg =560/100 = 5.60
1990 Per litre =280/10 =28
1996 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 1800
Wheat 150 kg P/quintal 110 1200
Jowar 100kg P/quintal 900 1000
Pulser 50 kg P/quintal 2500 3000
Suger 20 kg P/quintal 1300 1400
Oil 20 liter Per40liter 1000 1200
Fire wood 40 mounds Per mounds 30 50
Dal 10 kg Per kg 20 25
Kerosin oil 10 liter Per 20liter 60 80
Elolt 50 meter Per meter 10 15
SOLUTION Calculation for index number
Quantity
Commoditie 2002 2003 (P1/Po)
used Unit P1qo Poqo V PV
s Po P1 x 100=P
In 2002qo
Rice 200kg P/kg 15 18 3600 3000 120 3000 360000
Wheat 150kg P/kg 11 12 1800 1650 109 1650 180000
Jower 100kg P/kg 9 10 1000 900 111 900 100000
Pulses 50kg P/kg 25 30 1500 1250 120 1250 150000
Sugar 20kg P/lt 13 14 280 260 107.6 260 28000
Oil 20lit P/d 25 30 600 500 120 500 60000
Firewood 40md P/mt 30 50 2000 1200 166.7 1200 20000
Dal 10kg P/kg 20 25 250 200 125 200 25000
Kerosine oil 10lit P/lt 3 4 40 30 133 30 4000
Cloth 50mts P/mt 10 15 750 500 150 500 75000
11820 9490 9490 1182000
∑P1qo ∑Poqo ∑V ∑PV
177
15.
I. Aggregate expenditure Method
C.L.I = ∑P1qo x 100 = 11820 x 100 = 124.55
∑P0q0 9490
II FBM
C L I = ∑PV = 1182000 = 124.55
∑V 9490
ILLUSTRATION =12
An enquiry into the budgets of middle class families in Bungler gave the following information.
What changes in the cost of living index of 2001 have taken place as compared to 1999.
Expenses Food Rent Clothing Fuel Miscellaneous
Percentage 1991 35 15 20 10 20
Price relative 2001 116 120 125 125 150
SOLUTION
Items of expenditure Weights V P PV
Food 35 116 4060
Rent 15 120 1800
Clothing 20 125 2500
Fuel 10 125 1250
Miscellany 20 150 3000
100 636 126.10
Note:- 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 CPI
4. What do you mean by TRT and FRT? Show how Fisher’s formula satisfies these tests
5. 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 Problems
Problem 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.
1999 2001
Commodities
Price Rs Value Rs Price Rs Value Rs
A 30 600 40 1000
B 60 1500 120 2400
C 25 450 40 800
D 15 150 25 200
E 30 900 50 1600
Note = quantity = price, [ Answer FIN = 172.07]
178
16.
Problem No:2
Construct 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 12
B 25 6-00 7.50
C 10 5-00 5-25
D 20 48-00 52.00
E 65 15-00 16-50
F 30 9-00 27-00
Answers: A E M = CPI = 124.5
F B M = CPI = 124.5
Problem 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 Misecll
Particular % age 35% 10% 20% 15% 20%
Price 2001 150 25 75 30 40
Price 2002 145 23 65 30 45
[Answers CLI= 97.57]
Problem No = 4
Calculate Fisher’s index from the following data and show that it satisfies both TRT And FRT.
Commodities A B C D E
Base year – price in Rs 10 8 20 18 35
Base year – value in Rs 200 108 160 144 280
Current year – value in Rs 300 220 250 140 300
Current 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 = 5
The 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 48
Fuel and lighting 220 10
Clothing 230 8
Rent 160 12
Misee% 190 15
Answer C L I = 276 .41
C L I = 25706/93 = 276.41
Problem =6
Calculate 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 2004
Rice Per Kg 20 Kg 1-00 2-00
Wheat Per Kg 50 Kg 0.60 1-10
Oil Per Kg 10 Kg 2.00 4-00
Ghee Per Kg 500 Kg 8.00 14-00
Sugar Per Kg 5Kg 1.00 1-80
Cloth Per miter 40 Miter 2.00 3-75
House rent - One house 4.00 75-00
Answer :- C L I = 376/199 X 100 = 188.94
179
17.
Problem = 07
From the following data ,construct price index, using
a. Laspeyre’s Method
b. Paasche’s Method
c. Fishers Ideal index number
Commodities Base year Current year Base year Current year
Price Price in Rs Quantity Quantity
L 12 20 100 120
M 4 4 200 240
N 8 12 120 150
O 20 24 60 50
[Answers:- LIN = 136.5, PIN = 138.26 FIN = 137.4]
Problem No:- 8
Calculate 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 10 560
B 2 200 6 240
C 4 240 6 360
D 10 300 12 288
E 8 320 9 432
[Answer FIN = 152.9%]
Problem No=9
From the following data calculate the price index number using
a. Laspeyre’s method
b. Fisher’s ideal index Number
c. Also show how it satisfies time & Factor Reversal Test.
Commodities
A B C D E
Price Value Price Value Price Value Price Value Price Value
Year
Rs Rs Rs Rs Rs Rs Rs Rs Rs Rs
2001 16 1280 17 1700 16 960 19 760 15 900
2002 19 1710 22 3080 23 1610 21 1050 16 1280
[Answer LIN=123.2 FIN = 123.1]
Problem No:10
From the following data, construct cost of living index, by using
1. Aggregate Expenditure Method.
2. Family Budget Method.
Quantity used in Base year Current year
Commodities
Base year Price Rs Price In Rs
Wheat 4 8 14
Rice 2 15 21
Dal 1 10 14
Oil 5 20 30
Ghee 3 6 12
Cereals 1 7 14
Vegetables 2 5 15
[Answers: L I N = 165.21, F B M I N = 165.21]
180
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