Methods of Construction of Indices and Test of Adequacy
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Methods
of Construction of Indices and Test of Adequacy
1. Methods of Construction of Index
Numbers
There
are different ways of construction of index numbers. In general, construction
of index number if further available for the division in two parts: Simple and
Weighted. Furthermore, the simple method is classified into simple aggregative
and simple relative. Similarly, the weighted method is classified into weighted
aggregative and weight average of relative. The following chart illustrates the
various methods.
v UNWEIGHTED OR SIMPLE INDEX NUMBER
An
unweighted or simple price index number measures the percentage change in price
of a single item or a group of items between two periods of time. In unweighted
index number, all the values taken for study are of equal importance. There are
two methods in this category:
1) Simple
Aggregative Method
2) Simple
Average of Price Relatives Method
Simple
aggregative method:
Under this
method the prices of different items of current year are added and the total is
divided by the sum of prices of the base year items and multiplied by 100. The
following formula is used:
P01 =
∑p1 = Aggregate of Prices in
Current year
∑p0 = aggregate of Prices in Base year
P01 = Price Index number
Example 1. Construct the
Price Index Number for the year 1997, from the following information taking 1996
as base y ear.
|
Commodity |
Price in
2022 (Nu) |
Price in
2013 (Nu) |
|
A |
50 |
80 |
|
B |
40 |
60 |
|
C |
10 |
20 |
|
D |
5 |
10 |
|
E |
2 |
8 |
Methods
of Construction of Indices and Test of Adequacy
1. Methods of Construction of Index
Numbers
There
are different ways of construction of index numbers. In general, construction
of index number if further available for the division in two parts: Simple and
Weighted. Furthermore, the simple method is classified into simple aggregative
and simple relative. Similarly, the weighted method is classified into weighted
aggregative and weight average of relative. The following chart illustrates the
various methods.
v UNWEIGHTED OR SIMPLE INDEX NUMBER
An
unweighted or simple price index number measures the percentage change in price
of a single item or a group of items between two periods of time. In unweighted
index number, all the values taken for study are of equal importance. There are
two methods in this category:
1) Simple
Aggregative Method
2) Simple
Average of Price Relatives Method
Simple
aggregative method:
Under this
method the prices of different items of current year are added and the total is
divided by the sum of prices of the base year items and multiplied by 100. The
following formula is used:
P01 =
∑p1 = Aggregate of Prices in
Current year
∑p0 = aggregate of Prices in Base year
P01 = Price Index number
Example 1. Construct the
Price Index Number for the year 1997, from the following information taking 1996
as base y ear.
|
Commodity |
Price in
2022 (Nu) |
Price in
2013 (Nu) |
|
A |
50 |
80 |
|
B |
40 |
60 |
|
C |
10 |
20 |
|
D |
5 |
10 |
|
E |
2 |
8 |
Methods
of Construction of Indices and Test of Adequacy
1. Methods of Construction of Index
Numbers
There
are different ways of construction of index numbers. In general, construction
of index number if further available for the division in two parts: Simple and
Weighted. Furthermore, the simple method is classified into simple aggregative
and simple relative. Similarly, the weighted method is classified into weighted
aggregative and weight average of relative. The following chart illustrates the
various methods.
v UNWEIGHTED OR SIMPLE INDEX NUMBER
An
unweighted or simple price index number measures the percentage change in price
of a single item or a group of items between two periods of time. In unweighted
index number, all the values taken for study are of equal importance. There are
two methods in this category:
1) Simple
Aggregative Method
2) Simple
Average of Price Relatives Method
Simple
aggregative method:
Under this
method the prices of different items of current year are added and the total is
divided by the sum of prices of the base year items and multiplied by 100. The
following formula is used:
P01 =
∑p1 = Aggregate of Prices in
Current year
∑p0 = aggregate of Prices in Base year
P01 = Price Index number
Example 1. Construct the
Price Index Number for the year 1997, from the following information taking 1996
as base y ear.
|
Commodity |
Price in
2022 (Nu) |
Price in
2013 (Nu) |
|
A |
50 |
80 |
|
B |
40 |
60 |
|
C |
10 |
20 |
|
D |
5 |
10 |
|
E |
2 |
8 |
Solution: Construction of Price Index:
|
Commodity |
Price in
2013 (Nu) |
Price in
2022 (Nu) |
|
A |
50 |
80 |
|
B |
40 |
60 |
|
C |
10 |
20 |
|
D |
5 |
10 |
|
E |
2 |
8 |
|
|
∑p0 = 107 |
∑p1 = 178 |
P01 =
=
=
166.36%
Price Index in 2022, when compared to
2013 has fallen by 66.36%.
Simple Average of Price Relative
Method
In this method, first of all, the price
relatives of the commodities or items are found out. To compute the price
relatives, price in current year (p1) is divided by the price in
base year (p0) and then, the quotient is multiplied with 100. In
terms of formula:
Price Relative = 
Or
P = 
After this,
using Arithmetic average and Geometric average, we find the average of price
relatives.
(i)
When Arithmetic mean is
used, then the following formula is used:
P01 = 
where, N = number of items or commodities.
(ii)
When Geometric mean is
used, then the following formula is used:
P01 =
antilog 
Example 2. Compute price index
number by simple average of price relatives method using arithmetic mean and
geometric mean.
|
Item |
Price in
2010 (Nu) |
Price in
2011 (Nu) |
|
A |
6 |
10 |
|
B |
2 |
2 |
|
C |
4 |
6 |
|
D |
10 |
12 |
|
E |
8 |
12 |
Solution: Calculation of price index number by simple average of price
relatives:
|
Item |
Price in
2010 (Nu) |
Price in
2011 (Nu) |
P = |
|
|
A |
6 |
10 |
166.7 |
2.2201 |
|
B |
2 |
2 |
100 |
2 |
|
C |
4 |
6 |
150 |
2.1761 |
|
D |
10 |
12 |
120 |
2.0792 |
|
E |
8 |
12 |
150 |
2.1761 |
|
|
|
|
∑p = 686.7 |
∑ |
(i)
Price relative index number based on arithmetic mean:
P01 =
= 
= 
= 137.34
(ii)
Price relative index number based on geometric mean:
P01 = antilog or P01 = antilog
= antilog
= antilog (2.13303)
= 134.9
Hence,
the price index numbers based on arithmetic mean and geometric mean for the
year 2011 are 137.34 and 134.9 respectively.
v WEIGHTED INDEX NUMBERS
In computing weighted Index Numbers, the
weights are assigned to the items to bring out their economic importance.
Generally quanties consumed or values are used as weights. Weighted index
numbers are also of two types:
I.
Weighted Aggregative
II.
Weighted Average of Price
Relatives
Weighted
Aggregate Index Numbers
In
this method price of each commodity is weighted by the quantity sale either in
the base year or in the current year. There are various methods of assigning
weights and thus there are many methods of constructing index numbers. Some of
the important formulae used under these methods are:
a.
Laspeyre’s Index (P01L)
b.
Paasche’s Index (P01P)
c.
Dorbish and Bowley’s Index (P01DB)
d.
Fisher’s Ideal Index (P01F)
e.
Marshall-Edgeworth Index (P01Em)
f.
Kelly’s Index (P01K)
a) Laspeyre’s method
The base period quantities
are taken as weights the index is
P01L =
b) Paasche’s method
The current year quantities are taken as a
weight. In this method, we use continuously revised weights and thus this
method is not frequently used when the number of commodities is large. The
Index is
P01P =
c)
Dorbish
and Bowley’s method
In order in take into account the impact
of both the base and current year, we make use of simple arithmetic mean of
Laspeyre’s and Paasche’s formula. The index is
P01DB
=
= 
d)
Fisher’s
ideal method
It is the
geometric mean of Laspeyre’s Index and Paasche’s Index, given by:
P01F = 
= 
e) Marshall-Edgeworth
method
In this
method also the current year as well as base year prices and quantities are
considered. The Index is
P01ME
=
=
f) Kelly’s method
The Kelly’s Index is
P01K = 
Where, q refers
to quantity of some period, not necessarily of the mean of the base year and
current year. It is also possible to use average quantity of two or more years
as weights. This method is known as fixed weight aggregative index.
Example 3. Construct weighted
aggregate index numbers of price from the following data by applying
1.
Laspeyre’s method
2.
Paasche’s method
3.
Dorbish and Bowley’s method
4.
Fisher’s ideal method
5.
Marshall-Edgeworth method
|
Commodity |
2016 |
2017 |
||
|
Price |
Quantity |
Price |
Quantity |
|
|
A |
2 |
8 |
4 |
6 |
|
B |
5 |
10 |
6 |
5 |
|
C |
4 |
14 |
5 |
10 |
|
D |
2 |
19 |
2 |
13 |
Solution: calculation of various
indices
|
Commodity |
2016 |
2017 |
p1q0 |
p0q0 |
p1q1 |
p0q1 |
||
|
Price (p0) |
Quantity (q0) |
Price (p1) |
Quantity (q1) |
|||||
|
A |
2 |
8 |
4 |
6 |
32 |
16 |
24 |
12 |
|
B |
5 |
10 |
6 |
5 |
60 |
50 |
30 |
25 |
|
C |
4 |
14 |
5 |
10 |
70 |
56 |
50 |
40 |
|
D |
2 |
19 |
2 |
13 |
38 |
38 |
26 |
26 |
|
|
|
|
|
|
∑p1q0=200 |
∑p0q0=160 |
∑p1q1=130 |
∑p0q1=103 |
(1) Laspeyre’s method
P01L = 
= 
= 125
(2) Paasche’s method
P01P = 
= 
= 126.21
(3) Dorbish and Bowley’s method
P01DB
= 
= 
= 125.6
(4)
Fisher’s Ideal index
P01F = 
= 
= 125.61
(5)
Marshall-Edgeworth method
P01ME
= 
= 
= 
= 125.48
|
Commodity |
Quantity |
Price |
|
|
2007 |
2010 |
||
|
X |
25 |
3 |
4 |
|
Y |
12 |
5 |
7 |
|
Z |
10 |
6 |
5 |
Example 4. Calculate a suitable price index form the following
data.
Solution:
In this problem, the
quantities for both current year and base year are same. Hence, we can conclude
Kelly’s Index price number.
|
Commodity |
Quantity (q) |
Price |
p0q |
p1q |
|
|
2007 (p0) |
2010 (p1) |
||||
|
X |
25 |
3 |
4 |
75 |
100 |
|
Y |
12 |
5 |
7 |
60 |
84 |
|
Z |
10 |
6 |
5 |
60 |
50 |
|
|
|
|
|
∑p0q=195 |
∑p1q=234 |
P01K
= 
= 
= 120
Weighted
Average of Price Relative Method
The price relatives for
the current year are calculated on the basis of the base year prices. These
price relative are multiplied by the respective weights of the items. These
products are added up and then divided by the sum of weights.
(ii)
Price relative index number based on geometric mean:
P01 = antilog or P01 = antilog
= antilog
= antilog (2.13303)
= 134.9
Hence,
the price index numbers based on arithmetic mean and geometric mean for the
year 2011 are 137.34 and 134.9 respectively.
v WEIGHTED INDEX NUMBERS
In computing weighted Index Numbers, the
weights are assigned to the items to bring out their economic importance.
Generally quanties consumed or values are used as weights. Weighted index
numbers are also of two types:
I.
Weighted Aggregative
II.
Weighted Average of Price
Relatives
Weighted
Aggregate Index Numbers
In
this method price of each commodity is weighted by the quantity sale either in
the base year or in the current year. There are various methods of assigning
weights and thus there are many methods of constructing index numbers. Some of
the important formulae used under these methods are:
a.
Laspeyre’s Index (P01L)
b.
Paasche’s Index (P01P)
c.
Dorbish and Bowley’s Index (P01DB)
d.
Fisher’s Ideal Index (P01F)
e.
Marshall-Edgeworth Index (P01Em)
f.
Kelly’s Index (P01K)
a) Laspeyre’s method
The base period quantities
are taken as weights the index is
P01L =
b) Paasche’s method
The current year quantities are taken as a
weight. In this method, we use continuously revised weights and thus this
method is not frequently used when the number of commodities is large. The
Index is
P01P =
c)
Dorbish
and Bowley’s method
In order in take into account the impact
of both the base and current year, we make use of simple arithmetic mean of
Laspeyre’s and Paasche’s formula. The index is
P01DB
=
=
d)
Fisher’s
ideal method
It is the
geometric mean of Laspeyre’s Index and Paasche’s Index, given by:
P01F =
=
e) Marshall-Edgeworth
method
In this
method also the current year as well as base year prices and quantities are
considered. The Index is
P01ME
=
=
f) Kelly’s method
The Kelly’s Index is
P01K =
Where, q refers
to quantity of some period, not necessarily of the mean of the base year and
current year. It is also possible to use average quantity of two or more years
as weights. This method is known as fixed weight aggregative index.
Example 3. Construct weighted
aggregate index numbers of price from the following data by applying
1.
Laspeyre’s method
2.
Paasche’s method
3.
Dorbish and Bowley’s method
4.
Fisher’s ideal method
5.
Marshall-Edgeworth method
|
Commodity |
2016 |
2017 |
||
|
Price |
Quantity |
Price |
Quantity |
|
|
A |
2 |
8 |
4 |
6 |
|
B |
5 |
10 |
6 |
5 |
|
C |
4 |
14 |
5 |
10 |
|
D |
2 |
19 |
2 |
13 |
Solution: calculation of various
indices
|
Commodity |
2016 |
2017 |
p1q0 |
p0q0 |
p1q1 |
p0q1 |
||
|
Price (p0) |
Quantity (q0) |
Price (p1) |
Quantity (q1) |
|||||
|
A |
2 |
8 |
4 |
6 |
32 |
16 |
24 |
12 |
|
B |
5 |
10 |
6 |
5 |
60 |
50 |
30 |
25 |
|
C |
4 |
14 |
5 |
10 |
70 |
56 |
50 |
40 |
|
D |
2 |
19 |
2 |
13 |
38 |
38 |
26 |
26 |
|
|
|
|
|
|
∑p1q0=200 |
∑p0q0=160 |
∑p1q1=130 |
∑p0q1=103 |
(1) Laspeyre’s method
P01L =
=
= 125
(2) Paasche’s method
P01P =
=
= 126.21
(3) Dorbish and Bowley’s method
P01DB
=
=
= 125.6
(4)
Fisher’s Ideal index
P01F =
=
= 125.61
(5)
Marshall-Edgeworth method
P01ME
=
=
=
= 125.48
|
Commodity |
Quantity |
Price |
|
|
2007 |
2010 |
||
|
X |
25 |
3 |
4 |
|
Y |
12 |
5 |
7 |
|
Z |
10 |
6 |
5 |
Example 4. Calculate a suitable price index form the following
data.
Solution:
In this problem, the
quantities for both current year and base year are same. Hence, we can conclude
Kelly’s Index price number.
|
Commodity |
Quantity (q) |
Price |
p0q |
p1q |
|
|
2007 (p0) |
2010 (p1) |
||||
|
X |
25 |
3 |
4 |
75 |
100 |
|
Y |
12 |
5 |
7 |
60 |
84 |
|
Z |
10 |
6 |
5 |
60 |
50 |
|
|
|
|
|
∑p0q=195 |
∑p1q=234 |
P01K
=
=
= 120
Weighted
Average of Price Relative Method
The price relatives for
the current year are calculated on the basis of the base year prices. These
price relative are multiplied by the respective weights of the items. These
products are added up and then divided by the sum of weights.
P01 =
Where, p = is the price relative index
w = is value weights
Here also, we may use
either Arithmetic mean or Geometric mean for the purpose of averaging weighted
price relatives.
i.
The weighted average price relatives using
arithmetic mean:
P01
= = P01 =
ii.
The weight average price relatives using
geometric mean:
P01
= antilog
Example 5. Compute price
index for the following data by applying weighted average of price relatives
method using (i) Arithmetic mean and (ii) Geometric mean.
|
Commodity |
Quantity |
Price (Nu) |
|
|
2011 |
2019 |
||
|
X |
20 |
3 |
4 |
|
Y |
40 |
1.5 |
1.6 |
|
Z |
10 |
1 |
1.5 |
Solution:
|
item |
Quantity (q0) |
Price (Nu) |
W (p0q0) |
P
|
Log p |
pw |
w log p |
|
|
2011
(p0) |
2019 (p1) |
|||||||
|
X |
20 |
3 |
4 |
60 |
133.3 |
2.1249 |
8000 |
127.494 |
|
Y |
40 |
1.5 |
1.6 |
60 |
106.7 |
2.0282 |
6400 |
121.692 |
|
Z |
10 |
1 |
1.5 |
10 |
150 |
2.1761 |
1500 |
21.761 |
|
|
|
|
|
∑w=130 |
|
|
∑pw=15900 |
∑wlogp=270.947 |
(i)
Computation for the weighted average of
price relatives using arithmetic mean.
P01
=
=
= 122.31
This
means that there has been a 22.31% increase in prices over the base year.
(ii)
Index number using geometric mean of
price relatives is:
P01
= antilog
= antilog
= antilog (2.084)
= 121.3
This
means that there has been a 21.3% increase in prices over base year.
TESTS
OF ADEQUANCY FOR CONSTRUCTION OF INDEX NUMBER
1.
Method
of Constructing Test of Adequacy
The following tests can be applied to
find out the adequacy of an index number.
1)
Unite
Test
Formula
for construction of index numbers should not be affected by the unit in which
prices or quantities have been quoted. For example, in the group of
commodities, while the price of wheat might be in Kgs., that of vegetable oil
may be quoted in per liter and toilet soap may be per unit.
X Not satisfied by Simple (unweighted) Aggregative
index formula
(As units play an important
part in determining value of in the index.)
All
other formulae discussed above satisfy this test.
2)
Time Reversal Test
The
time reversal test is used to test whether a give method will work both
backwards and forwards with respect to time.
Forward
Backward
Formula is such that it gives the same
ratio between one point of comparison and another no matter which of the two is
taken as base.
The
time reversal test may be stated more precisely as follows-
· If
the time subscripts of a price (or quantity) index number formula be
interchanged, the resulting price (or quantity) formula should be reciprocal of
the original formula.
i.e. if p0 represents price of
year 2011 and p1 represents
price at year 2012 i.e.
should be equal to 
Symbolically, the following relation
should be satisfied
, omitting the factor
100 from both the indices.
Where, p01 is index for current year ‘1’ based on base year
‘0’
P10 is index for
year ‘0’ based on year ‘1’.
· The
methods which satisfy the following test are:
1. Simple
aggregate index
2. Simple
geometric mean of price relative
3. Weighted
geometric mean of price relative with fixed weights
4. Kelly’s
fixed weight formula
5. Fisher’s
ideal formula
6. Marshall-Edgeworth
formula
Proof:
Fisher’s
Price Number:
P01 = 
(omitting the factor 100)
Interchange
, p0 p1
p1 p0
q0 q1
q1 q0
P10 =
= 
= 1
Hence the test is satisfied.
· This test is not satisfied by Laspeyre’s
method and Paasche’s method as
(i) When
Laspeyre’s Method is used
(ii) When
Paasche’s Method is used
3) Factor
Reversal Test
Product of a price index and the quantity
index should be equal to the corresponding value index.
If, Price of commodity is doubled
Quantity of commodity is tripled
Total
change in value should be six times
X Not satisfied by rest
of the method
Consider the Laspeyre’s formula of price index
Consider the quantity index by interchange
‘p’ with ‘p’ and ‘q’ with ‘p
Now 
This test is not met.
This test is only satisfied by Fisher’s ideal index
Proof:
P01 =
Changing ‘p’ to ‘q’ and ‘q’ to ‘p’ we got
Q10 = 
Thus Fisher’s Ideal Index satisfies Factor
Reversal Test.
4) Circular
Test
Circular
test is an extension of time reversal test for more than two periods and is
based on shift ability of the base period.
It is an
extension of time reversal test (which is of two years.)
If we consider 3 years
2012 with base year
2011 P01
2011 with base year 2010 P12
Then we ought to get same result if we had directly calculated index for
2012 with the base year 2010 P20
Symbolically,
This test is satisfied only by the following index numbers
formulas-
1)
Simple
aggregative index
2)
Kelly’s
fixed base method
X The test is not satisfied by Fisher’s Ideal
index’
Example 6. Form
the following data proves that Fisher’s Ideal index satisfies time reversal
test and factor reversal test.
|
Item |
2010 |
2011 |
||
|
Price |
Quantity |
Price |
Quantity |
|
|
A |
6 |
50 |
10 |
60 |
|
B |
2 |
100 |
2 |
120 |
|
C |
4 |
60 |
6 |
60 |
Solution:
|
item |
2010 |
2011 |
p1q0 |
p0q0 |
p1q1 |
p0q1 |
||
|
Price(p0) |
Quantity(q0) |
Price(p1) |
Quantity(q1) |
|||||
|
A |
6 |
50 |
10 |
60 |
500 |
300 |
600 |
360 |
|
B |
2 |
100 |
2 |
120 |
200 |
200 |
240 |
240 |
|
C |
4 |
60 |
6 |
60 |
360 |
240 |
360 |
240 |
|
|
|
|
|
|
∑p1q0=1060 |
∑p0q0=740 |
∑p1q1=1200 |
∑p0q1=840 |
1.
Calculation of Fisher’s Ideal index to
prove time reversal time
Fisher’s
price index
P01 = 
Fisher’s price index
P10 = 
Time reveral test is satisfied when
= 
= 
= = 1
Since
hence Fisher’s ideal index is satisfies time
reversal test.
2.
Calculation of Fisher’s ideal index to
prove factor reversal test
Fisher’s
quantity index
Q10 = 
Factor reversal test is satisfied when
=
=
=
Value index = 
=
Hence Fisher’s ideal index
satisfies factor reversal test
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