Index number

Index Number
When we go to the market after a long gap, we find that
the prices of most commodities have changed. Some
items have become costlier, while others have become
cheaper.
It is bewildering to us some times, but it is fact.

We see that there are changes in prices of
commodities, income or salaries of consumers,
share prices, industrial outcome, there is no
uniformity is seen in it. Description of the
individual rates of change is difficult to
understand.
But with the help of a statistical device, change in
magnitude of a group of related variables can be
measured is called Index number.

India ranks 46th on Global Innovation
Index 2021. Did you know how such
ranks are calculated? Our country’s
rank is as much of concern as what
index numbers are because such
calculations are done using index
numbers. Let us dive deep into the sea
of index numbers.

Wholesale price
indexes for United
States, Great Britain,
Germany, and France,
1790–1940.

Price Relative
Price relative is the ratio of the price of
a specific product in one period to the
price of the same product in some other
period.

This can be illustrated through calculation of
simple aggregative price Index:
INDEX NUMBER
An index number is a statistical device for measuring changes
in the magnitude of a group of related variables. It represents
the general trend of diverging ratios, from which it is calculated.
It is a measure of the average change in a group of related
variables over two different situations.
The comparison may be between like categories such as persons, schools,
hospitals etc. An index number also measures changes in the value of the
variables such as prices of specified list of commodities, volume of
production in different sectors of an industry, production of various
agricultural crops, cost of living etc.

Conventionally, index numbers are expressed in terms of percentage. Of the
two periods, the period with which the comparison is to be made, is known
as the base period. The value in the base period is given the index number
100.
If you want to know how much the price has changed in 2005 from the level
in 1990, then 1990 becomes the base period. The index number of any
period is in proportion with it. Thus an index number of 250 indicates that
the value is two and half times that of the base period.
Price index numbers measure and permit comparison of the prices of
certain goods. Quantity index numbers measure the changes in the
physical volume of production, construction or employment. Though price
index numbers are more widely used, a production index is also an
important indicator of the level of the output in the economy.

CONSTRUCTION OF AN INDEX NUMBER
In the following sections, the principles of constructing an index number
will be illustrated through price index numbers.
Let us look at the following example: Calculation of simple aggregative
price index
As we see in this example, the percentage changes
are different for every commodity. If the percentage
changes were the same for all four items, a single
measure would have been sufficient to describe the
change.
However, the percentage changes differ and
reporting the percentage change for every item will
be confusing. It happens when the number of
commodities is large, which is common in any real
market situation. A price index represents these
changes by a single numerical measure.

There are two methods of constructing an index number. It can be
computed by the aggregative method and by the method of averaging
relatives.
Example:
Using the data from previous example,
the simple aggregative price index is:

The price index number using price
relatives is

The reason is that the units of measurement of
prices of various commodities are not the
same. It is unweighted, because the relative
importance of the items has not been properly
reflected.
The items are treated as having equal
importance or weight. But what happens in
reality? In reality the items purchased differ in
order of importance. Food items occupy a large
proportion of our expenditure.
In that case an equal rise in the price of an item
with large weight and that of an item with low
weight will have different implications for the
overall change in the price index.
Po = Base Period Price
qo = Base Period Quantity
P1 = Current Period Price
q1 = Current Period Quantity
Simple Aggregative Method is of limited use?

Here weights are the quantity, to construct a weighted aggregative
index, a well-specified basket of commodities is taken and its worth
each year is calculated. It thus measures the changing value of a
fixed aggregate of goods. Since the total value changes with a fixed
basket, the change is due to price change. Various methods of
calculating a weighted aggregative index use different baskets with
respect to time.

Po = Base Period Price,
P1 = Current Period Price,
Using base period quantities as
weights, the price is said to have risen
by 35.3 percent.
A weighted aggregative price index
using base period quantities as weights,
is also known as Laspeyre’s price index.
(1834-1913)
It was given by a German economist,
statistician Ernst Louis Étienne
Laspeyres, Professor at University of
Giessen in 1871.

Since the current period quantities
differ from the base period quantities,
the index number using current period
weights gives a different value of the
index number. It is known as Paasche’s
price index:
It is given by formula as below:
Po = Base Period Price,
P1 = Current Period Price,
Paasche index developed by
German economist Hermann
Paasche in 1874.
Using current period weights, the price
is said to have risen by 32.1 per cent.

= 133.69
The Paasche value being less
than the Laspeyres indicates
usage has increased faster in
the lower period sector.

Method of Averaging relatives
When there is only one commodity, the
price index is the ratio of the price of the
commodity in the current period to that
in the base period.
The method of averaging relatives takes
the average of these relatives when there
are many commodities. The price index
number using price relatives is given as:
where P1 and Po indicate the price of the
ith commodity in the current period and
base period respectively.
The ratio (P1/P0) × 100 is also referred to as
price relative of the commodity, n stands
for the number of commodities.
The price index number using price
relatives is calculated as:
Thus, the prices of the commodities
have risen by 49 per cent.

Quantity index numbers
Quantity index numbers measure the
change in the quantity or volume of
goods sold, consumed or produced
during a given time period. Hence it is a
measure of relative changes over a
period of time in the quantities of a
particular set of goods.
Just like price index numbers and
value index umbers, there are also
two types of quantity index numbers,
namely
• Unweighted Quantity Indices
• Weighted Quantity Indices

Unweighted Index: Simple Aggregate
Method
Here we do a simple and direct comparison of
the aggregate quantities of the current year,
with those of the previous year.
We express this index number as a
percentage. No weights are assigned, it is the
simplest calculation.
The formula is as follows,
where, Q1 is the quantity of the current year,
and Q0 is the quantity of the previous year,
Unweighted Index: Simple Average of Quantity
Method
In this method, we take the aggregate quantities
of the current year as a percentage of the
quantity of the base year. Then to obtain the
index number, we average this percentage
figure.
So the formula under this method is as follows,
where N is the total number of items

Weighted Index: Simple Aggregative Method
There are a few various methods for
calculating this index number. We will take a
look at some of the most important ones.
1] Laspeyres Method
In this method, the base price is taken as the
weight. We only use the price of the base year
(P0), not the current year. The formula is as
follows,
2] Paasche’s Method
Here, the current year price (P1) of the
commodity is taken as the weight.

Weighted Index : Weighted Average
of Relative Method
In this method, we use the arithmetic
mean for averaging the values. The
formula is a little more complex as
below,

The weighted price index is 156. The price
index has risen by 56 per cent. The values
of the unweighted price index and the
weighted price index differ, as they should.
The higher rise in the weighted index is
due to the doubling of the most important
item A

Today we have n numbers of index numbers for every sector to
measure the degree of economic changes over time and predicting the
future requirement.

Consumer price index (CPI), also known as the
cost of living index.
It measures the average change in retail prices. Consider
the statement that the CPI for industrial workers
(2001=100) is 277 in December 2014.
What does this statement mean?
It means that if the industrial worker was spending Rs
100 in 2001 for a typical basket of commodities, he needs
Rs 277 in December 2014 to be able to buy an identical
basket of commodities.
It is not necessary that he/she buys the basket. What is
important is whether he has the capability to buy it.

CONCLUSION
• Estimating index number enables us to calculate a single
measure of change of a large number of items.
• Index numbers can be calculated for price, quantity, volume, etc.
• It is also clear from the formulae that the index numbers need to
be interpreted carefully.
• The items to be included and the choice of the base period are
important.
• Index numbers are extremely important in policy making as is
evident by their various uses.

Index number

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