Index Numbers
1McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
Contents
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Definition
Classification
Methods of construction
Unweighted Indices
Weighted Indices
Tests of adequacy of indices formulae
Value Index number
Consumer Price Index number
Definition
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An index number measures the
relative change in price, quantity,
value, or some other item of interest
from one time period to another.
A simple index number measures the
relative change in one or more than
one variable.
According to Blair “Index numbers are
a specialized type of averages.”
“Index numbers are used to measure
the changes in some quantity which we
cannot observe directly”.
- Bowley
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©The McGraw-Hill Companies, Inc. 2008
Classification of index numbers
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Price index numbers: It measures the
changes in the prices of the commodities
produced, consumed or sold in a given period
with reference to the base period.
Quantity index numbers: These help to
measure and compare the changes in the
physical volume of goods produced, sold and
purchased in a given period compared to some
other given period.
Value index numbers: These indexes show
changes in the value of any commodity in a
given period in reference to the base period.
Classification (contd.)
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Consumer price index: These indexes
measure the average over time in the
prices paid by the consumers for a specific
group of goods and services.
Special purpose index numbers: These
indexes are framed for a special study
relating to a particular variable or aspect.
Methods of construction of index
numbers
Index
numbers
Simple or
unweighted
Simple
aggregative
method
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Simple Price
relative
method
Weighted
Weighted
aggregative
method
Weighted
price relative
method
Unweighted Indices
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Simple average - example
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Simple Aggregate Index – Example
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Weighted Indices
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When all commodities are not of equal importance.
We assign weight to each commodity relative to its
importance and index number computed from these
weights is called weighted index numbers.
Laspeyre’s Index Number: In this index
number the base year quantities are used as
weights, so it also called base year weighted index.
Weighted Indexes (contd.)
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Paasche’s Index Numbers: In this index
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Marshal-Edgeworth Index Number: In this
number, the current (given) year quantities are
used as weights, so it is also called current
year weighted index.
index number, the average of the base year and
current year quantities are used as weights. This
index number is proposed by two English
economists Marshal and Edgeworth.
Weighted Indexes (contd.)
Marshal-Edgeworth Index Number
(contd.):
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Fisher’s Ideal Index Numbers: Geometric
mean of Laspeyre’s and Paasche’s index numbers is
known as Fisher’s ideal index number. It is called
ideal because it satisfies the time reversal and
factor reversal test.
Weighted Indexes (contd.)
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Dorbish and Bowley’s index numbers: This
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Kelley’s index numbers: Kelly believes that a
method takes into account base as well as current
period for the construction of index numbers. It
is the average of Laspeyer’s and Paasche’s
method.
ratio of aggregates with selected weights (not
necessarily of base year or current year) gives
the base index number.
Laspeyres vs. Paasche Index
When is Laspeyres most appropriate &
when is Paasche the better choice?
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Laspeyre’s:
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Ernst Louis
Étienne Laspeyres
(1834 – 1913)
Advantages: Requires quantity data
from only the base period. This allows a
more meaningful comparison over time.
The changes in the index can be
attributed to changes in the price.
Disadvantages: Does not reflect
changes in buying patterns over time.
Also, it may overweight goods whose
prices increase.
Paasche’s:
Advantages: Because it uses
quantities from the current period,
it reflects current buying habits.
Hermann Paasche (1851–
1925)
Disadvantages: It requires quantity data for
the current year. Because different quantities
are used each year, it is impossible to attribute
changes in the index to changes in price alone.
It tends to overweight the goods whose prices
have declined. It requires the prices to be
recomputed each year.
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Fisher’s Ideal Index
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Laspeyres’ index tends to
overweight goods whose prices
have increased.
Sir Ronald Aylmer
Paasche’s index, on the other hand,
Fisher (1890 -1962)
tends to overweight goods whose prices have
gone down.
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Fisher’s ideal index was developed in an
attempt to offset these shortcomings.
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It is the geometric mean of the Laspeyres and
Paasche indexes.
Weighted Indexes - Example
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Weighted Indexes – Example (contd.)
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Weighted Indexes – Example (contd.)
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Time reversal test
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It is a test to determine whether a given method
will work both ways in time, forward and
backward. In the words of Fisher, “The test is
that the formula for calculating the index number
should be such that it will give the same ratio
between one point of comparison and the other, no
matter which of the two is taken as base.”
Symbolically, the following relation should be
satisfied: P01 X P10 = 1
Where P01 is the index for time “I” on time “0” as
base and P10 is the index for time “0” on time “I”
as base. If the product is not unity, there is said
to be a time bias in the method.
Time reversal test (contd.)
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The test is not satisfied by Laspeyres method and
the Paasche method as can be seen below:
When Laspeyres method is usedand the test is not satisfied.
When Paasche method is used-
and the test is not satisfied.
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Time reversal test (contd.)
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Let us now see how Fisher’s Ideal formula satisfies
the test.
Proof:
Changing time, i.e., 0 to 1 and 1 to 0.
Since P01 X P10 = 1, the Fisher’s ideal index satisfies
the test.
Factor reversal test
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Another test suggested by Fisher is known
as factor reversal test. It holds that the
product of a price index and the quantity
index should be equal to the corresponding
value index. In the words of Fisher, “Just as
each formula should permit the interchange
the prices and quantities without giving
inconsistent results, so it ought to permit the
interchange of the two times without giving
inconsistent results, i.e., the two results
multiplied together should give the true value
ratio.”
Factor reversal test (contd.)
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The factor reversal test is satisfied only by the
Fisher’s Ideal Index.
Proof:
Change p to q and q to p:
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Factor reversal test (contd.)
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Since
the factor reversal
test is satisfied by the Fisher’s Ideal
index. This means, of course, that the
formula serves equally well for
constructing indices of quantities as for
constructing indices of prices, the quantity
index being derived by interchanging p and
q in the ideal formulae.
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Circular test
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This test is just an extension of the time reversal
test. The test requires that if an index is
constructed for the year a on base year b, and for
the year b on base year c, we ought to get the same
result as if we calculated direct an index for a on
base year c without going through b as an
intermediary. The Laspeyres index does not satisfy
the test as can be seen from the following:
If the three years are 0, 1, 2, the index by
Laspeyres method will be:
Circular test (contd.)
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The product of all these is not equal to 1. Hence
the test is not satisfied. Similarly, it can be
shown that the Paasche’s index and Fisher’s index
do not satisfy the test.
The circular test is not met by the ideal index or
by any of weighted aggregative with changing
weights. This test is met by simple geometric
mean of price relatives and the weighted
aggregative fixed weights.
Value Index
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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.
Its formula is:
Value Index - Example
The prices and quantities sold at the Waleska
Clothing Emporium for various items of apparel for
May 2000 and May 2005 are:
What is the index of value for May 2005 using May 2000 as the base period?
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Value Index – Example (contd.)
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Consumer Price Index numbers
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Also known as cost of living index numbers, these
are generally intended to represent the average
change over time in the prices paid by the
ultimate consumer of a specified basket of goods
and services.
Uses :
1.Serves the basis for wage negotiations &
contracts.
2.Helps in formulation of wage policy, price policy,
etc.
3.Also measures varying purchasing power of
currency.
Thank you
33McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008