INDEX NUMBERS

An index number is a statistical device (measure) with a purpose of showing average changes in one or more related variables (like price or quantity) between two periods of time (say, between 1991 and 1996) or two places like cities (say, Delhi and Mumbai) or countries (say, India and Japan). For example, we may be interested in knowing which city of India out of Delhi, Mumbai, Kolkata and Chennai is the costliest or the cheapest in terms of price level. A tourist may be interested in knowing about the cost of living at different tourist places. Consumer price index number or cost of living index number helps in taking such decisions.

Different commodities are measured in different units. For example, rice and wheat are measured in kilograms, cloth in meters and milk in liters etc. Index numbers attempt some averages relating to commodities which are measured in different units. For purpose of comparison, we are interested in knowing relative changes. These relative changes are expressed in percentage terms.

Index numbers are the indicators of the various trends in an economy. Price index numbers indicate the position of prices whether they are rising or falling and at what rate. Similarly, index numbers regarding agricultural production indicates the trend of change whether it is rising or falling at what rate over a period of time. An index number is an economic data figure reflecting price or quantity compared with a standard or base value. The base usually equals 100 and the index number is usually expressed as 100 times the ratio to the base value. For example, if a commodity costs twice as much in 1970 as it did in 1960, its index number would be 200 relative to 1960. Index numbers are used especially to compare business activity, the cost of living, and employment.

An index number is specialized average. Index numbers may be simple or weighted depending on whether we assign equal importance to every commodities or different importance to different commodities according to the percentage of income spent on them or on the basis of some other criteria. In this chapter, we shall discuss both simple and weighted index numbers.

Uses of Index Numbers

The main uses of index numbers are given below:

Index numbers are used in the fields of commerce, meteorology, labour, industrial, etc.
The index numbers measure fluctuations during intervals of time, group differences of geographical position of degree etc.
They are used to compare the total variations in the prices of different commodities in which the unit of measurements differs with time and price etc.
They measure the purchasing power of money.
They are helpful in forecasting the future economic trends.
They are used in studying difference between the comparable categories of animals, persons or items.
Index numbers of industrial production are used to measure the changes in the level of industrial production in the country.
Index numbers of import prices and export prices are used to measure the changes in the trade of a country.

The index numbers are used to measure seasonal variations and cyclical variations in a time series.

TYPES OF SIMPLE INDEX NUMBERS

1) Price Relatives:

One of the simplest types of index numbers is a price relative. It is the ratio of the price of a single commodity in a given period or point of time to its price in another period or point of time, called the reference period or base period. If the prices for a period, instead of a point of time, are considered, then suitable price average for the period is taken and these prices are expressed in the same units. If and denote the price of a commodity during the base period or reference period (0) and the given period (1) then the price relative of the period 1 with respect to the base period 0 is defined by

Price relative in percentage (of period 1 with respect to 0)

We denote price relative in percentage or without percentage by .

Remark: In the formula of price relative we multiply p1 / p0 by 100 only to get a better expression. This expression is therefore called price relative in percentage.
Exercise 1: If the retail price of fine quality of rice in the year 1980 was Rs.3.75 and that for the year 1983 was Rs.4.50, then find the price relative.
Solution: Here the base period is 1980. The price of rice in the base period was Rs.3.75. Also in the given period 1983, the price of rice was Rs.4.50. Using the formula (19.1), the required price relative is

Exercise 2: The exchange rate of a US dollar was Rs.40.00 in July 2008 and was Rs.50.00 in February 2009. Find the price relative.
Solution: The price relative of a dollar in February 2009 (say period F) with respect to that in July 2008 (say period J) is given by

2) Quantity Relatives:
Another simple type of index number is a quantity relative. This is useful when we are interested in changes in quantum or volume of a commodity such as quantities of production or sale or consumption. Here the commodity is used in a more general sense. It may mean the volume of goods (in tonnes) carried by roadways or the volume of export to a country or import from a country. In such cases we consider quantity or volume relatives. If quantities or volumes are taken for a period instead of a point of time, a suitable average is to be taken and the quantities or volumes are to be expressed in the same units. If and denote the quantity or volume produced, consumed or transacted during the base period (0) and the given period (1) then quantity relative of the period 1 with respect to the base period 0 is defined by

Quantity relative in percentage (of period 1 with respect to 0)

We denote quantity relative in percentage or without percentage by q0 / 1 .
Exercise 3: The production of tea in Assam in the month of January 2009 was 3608 tonnes and that in February 2009 was 3700 tonnes. Find the quantity relative.
Solution: The quantity relative of tea in February 2009 (say period F) with respect to that in January 2009 (say period J) is given by

3) Value Relatives:
A value relative is another type of simple index number. It is usable when we wish to compare changes in the money value of the transaction, consumption or sale in two different periods or points of time. Multiplying the quantity by the price of the commodity produced, transacted or sold gives the total money value of the production, transaction or sale. If instead of point of time, period of time is considered, a suitable average is to be taken and is to be expressed in the same units.

Suppose p0 and q0 denote the price and quantity of the commodity during the base period (0) and p1 and q1 denote the corresponding price and quantity during a given period (1). Then the total value of the commodity during the base period is v0= p0q0. and the corresponding total value during the given period is v1= p1q1 . The value relative of the period 1 with respect to the base period 0 is defined by
Value relative in percentage (of period 1 with respect to 0)

We denote value relative in percentage or without percentage by.
Exercise 4: The production of tea in Assam in the month of January 2009 was 3608 tonnes and that in February 2009 was 3700 tonnes. The corresponding prices of tea were Rs.100.00 and Rs.120.00 per kilogram, respectively. Find the value relative.

PROPERTIES OF RELATIVES

The required property follows by multiplying the above three terms.
4) Modified circular or cyclic property: This property states that

PROPERTIES OF RELATIVES

In general we are interested in general price level rather than on the price level of any particular item. So we have to consider a number of items at the same time. If each commodity is of different qualities then all such qualities have also to be taken into consideration. We have to consider simultaneously a number of items or commodities together for measuring change in price or production.
The sources of data must be reliable and care should be taken in selecting the sources of data. For example, official publications as well as other reliable unofficial reports from producers and merchants should be considered for computing index for production. If retail food prices are being considered, then prices in stores, not in wholesale markets, are to be taken into consideration.

Now we shall discuss how to construct simple index numbers from a given set of data, like prices of certain commodities, production of certain commodities etc.
There are two methods of constructing simple index numbers. These are:

1) Simple aggregate method and
2) Method of simple average of relatives.
Remember that all items in a simple index number are assigned equal weights. These are unweighted index numbers.
1) Simple Aggregate Method: In this case each item is given equal weights. If we give an equal weight to each item it means the same thing, whether each item is given a weight or not. It is the simplest method of constructing index numbers.

1.1) Simple aggregate method to find price index numbers:
We use the following three steps to find price index number.

Solution: Represent prices of 2005 (current year) as p1 and that of 2000 (base year) as p0 and reconstruct the table as follows:

Exercise 6: The prices of three commodities A, B and C increased from Rs.50, Rs.7 and Rs.3 in 2005 to Rs.52, Rs.8 and Rs.5 respectively in 2006. Using the simple aggregate method, find by how much on an average the prices have increased?
Solution: We know construct the following table:

This shows that the price index of current year 2006 with respect to base year 2005 is 108.33%. Hence the price increase in one year is 108.33 – 100 = 8.33%.

1.2) Simple aggregate method to find quantity index numbers:
We use the following three steps to find quantity index number.
Step 1: Find the sum of the current year (that is, given period) quantity or volume produced, consumed or transacted of all the items included in the list. If 1 denote the current year, then find
Step 2: Find the sum of the base year quantity or volume produced, consumed or transacted of all the items included in the list. If 0 denote the base year, then find

Step 3: Use the formula

2) Method of Simple Average of Relatives: In this method, average of the relatives is obtained by using any one of the measures of central tendency. In particular, here we shall use arithmetic mean for averaging the relatives. We shall discuss briefly this method only for price relatives. Similar formulas on quantity index number and value index number can be obtained considering quantity and value relatives instead of price relative. There are few questions to find quantity and value index numbers in the section “Possible Questions”. You should work out those problems with the help of the problems on price index numbers which are solved below.

2.1) Method of Simple Average of Price Relatives:
We know that a price relative is nothing but the ratio of current year prices to those in the base year. Taking the arithmetic mean of the price relatives, we get the following formula for price index number:

where N stands for number of commodities included in the index numbers.
Steps to calculate price index numbers by Simple Average of Price Relatives Method:
Step 1: Find percentage price relative for each commodity, that is,

Step 2: Find the sum of these percentage price relatives, that is,

Exercise 7: Give the solution of Exercise using the Simple Average of Price Relatives.

Solution: We construct the following table:

Hence, the price index number of 2005 with respect to base year 2000 is:

Exercise 8: Give the solution of Exercise using the Simple Average of Price Relatives.

Solution: We first prepare the following table:

CONSTRUCTION OF WEIGHTED INDEX NUMBERS

In weighted index number each item is given weight according to the importance it occupies in the list. There are two groups of methods to calculate index number of this category: (a) Weighted aggregate method and (b) Weighted average of price relative methods. In this unit, we will study three types of weighted aggregate method. These methods are popularly known as (1) Laspeyre’s Method, (2) Paasche’s Method and (3) Fisher’s Method. All the methods are used to calculate weighted index numbers. The main difference between the Laspeyre’s Method and Paasche’s Method is that Laspeyre’s uses base year quantities of commodities as their relative weights, while Paasche’s uses current year quantities of commodities as their relative weights for preparing a price index.
Weighted Index Numbers

          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.


Paasche’s Index Number:
            In this index number, the current (given) year quantities are used as weights, so it is also called current year weighted index.


Fisher’s Ideal Index Number:
            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.



Marshal-Edgeworth Index Number:
            In this 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.



Example:
            Compute the weighted aggregative price index numbers for  with as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number (4) Marshal Edgeworth Index Number.

Commodity

Prices

Quantities

Solution:


Commodity

Prices

Quantity

Laspeyre’s Index Number

Paashe’s Index Number

Fisher’s Ideal Index Number


Marshal Edgeworth Index Number



(1) Laspeyre’s Method:
Laspeyre’s Price Index Number: It uses base year quantities as the weights. Accordingly, the formula is

Laspeyre’s Quantity Index Number: If we multiply the quantities by the base year price, then we get Laspeyre’s quantity index number. Accordingly the formula for Laspeyre’s quantity index number is:

Steps to Calculate Weighted Index Number by Laspeyre’s Method:
Step 1: Multiply current year price (p1) with the base year quantity (q0 ) to get p1q0 for each item/ commodity.

(2) Paasche’s Method:
Paasche’s Price Index Number: It uses current year quantities (q1) as the weights. Accordingly, the formula is

Paasche’s Quantity Index Number: It uses current year prices (p1) as the weights. Accordingly, the formula is

Exercise 10: Find Paasche’s price index number from the data given in Exercise
Solution: Represent prices and quantities in current year by p1 and q1 respectively and represent prices and quantities in base year by p0 and q0 respectively. Let us construct the following table.

Remark: You have noticed that the value of Laspeyre’s index number (107.38) and the value of Paasche’s index number (106.8) are different for the same data. The difference has arisen because of differences in weights.
(3) Fisher’s Method: The Fisher’s price index number is nothing but the geometric mean of the Laspeyre’s and Paasche’s price index numbers. Hence the formula for Fisher’s price index number is:

Similarly, the Fisher’s quantity index number is given by:

Exercise 11: Find Fisher’s price index number from the data given in Exercise.
Solution: We have already found that for the given data,

Quantity Volume Index Numbers

Price index numbers measure and permit comparison of the price of certain goods, quantity index number on te other hand measure the physical volume of production construction or employment though price indices volume of production construction or employment though pike indices are more widely used. Production indices are highly significant as indicator of the level of output in the economy or in parts of it.

In constructing quantity index numbers, the problems confronting the statistician are analogous to those involved in price indices. We measure changes in quantities and when we weight we use prices or values as weights quantity indices can be obtained easily by changing p to q and q to p in the various formulae discussed above.

Thus when Laspeyres method is used 01 Σ a₁ p₀/Σ q₀ p₀ x 100

When Paasche formula is used  q₀₁ = Σ q₁ p₁ / Σ q₀ p₁ x 100

When fisher formula is used

These formulae represent the quantity index in which the quantities of the different commodities are weighted by their prices. However, any other suitable weights can be used instead.

Illustration

From the following data compute a quantity index

Commodity 2008 Quantity 2009 Price in 2008

A 30 25 30

B 20 30 40

C 10 15 20

Solution

Computation of quantity index

Commodity q₀ q₁ p₀ q₁p₀ q₀p₀

A 30 25 30 750 900

B 20 30 40 1200 800

C 10 15 20 300 200

Σ q₁p₀ = 2,250 Σ q₀p₀ = 1,900

q₀₁ = Σ q₁ p₀/Σ q₀p₀ x 100 = 2,250 / 1,900 x 100 = 118. 42

Thus compared to 2008 the quantity index has gone up by 18.42 per cent in 2009,

Illustration:

Compute by fisher index formula the quantity index from the data given below:

Commodity Price Total value Price Total value

A 10 100 8 96

B 16 96 14 98

C 12 36 10 40

Solution: since we are given the value and the price we can obtain quantity figure by dividing value figures by price for each commodity we can then apply fisher’s ideal formula:

Computation of quantity index by fisher’s method

Commodity p₀ q₀ p₁ q₁ q₁p₀ q₀p₀ q₁p₁ q₀p₁

A 10 10 8 12 120 100 96 80

B 16 6 14 7 112 96 98 84

C 12 3 10 4 48 36 40 30

Σ q₁p₀ = 280 Σ q₀p₀ = 232 Σ q₁p₁ = 234 Σ q₀p₁ = 194

= √280/232 x 234 / 194 x 100

= √1.45574 x 100 = 1.2065 x 100 = 120.65

TEST OF
Adequacy Of Index Number

We have seen in section 14.5 that the relatives have certain important properties. What is true for an individual commodity should also true for a group of commodities. The index numbers as an aggregate relative should also satisfy the same set of properties. We shall examine the properties that a good index number should have.
1. Time Reversal Test or Property:

If the two periods, the base period and the reference period are interchanged, the product of the two index numbers should be unity. In other words, one should be the reciprocal of the other.

Consequently, Laspeyre’s price index number does not satisfy the time reversal property. Similarly, it can be seen that Laspeyre’s quantity index number also does not satisfy the time reversal property. Also, Paasche’s price and quantity index numbers do not satisfy this property.

Now consider the Fisher’s price index number. Interchanging the time subscripts we find another index number. The product of the two is given by

which is equal to 1. Hence Fisher’s price index number satisfies the time reversal property.
2. Factor Reversal Test or Property:
If the two factors and in a price index formula are interchanged, so that a quantity index number is obtained, then the product of the two index numbers should give the true value ratio

In other words, the price index number multiplied by the corresponding quantity index number should give the true ratio of value in the given year (1) to the value in the base year (0). This property holds for a single commodity, since

but it does not hold for most of the index numbers. For example,

Now consider the Fisher’s index number. We have from the formula's (1) and (2),

This proves that Fisher’s index number also satisfy factor reversal test. Fisher’s index number is one of the rare index numbers which satisfy both the time reversal and factor reversal tests. So it is called “Fisher’s Ideal Index Number”.

2. Circular Test:
This test is based on the shifting of the base period. If denotes the index number for the given period with respect to the base period then this test requires that

Remark: As we have seen in section 14.5, all the relatives satisfy the circular property.
Remark: Verify that none of the weighted index numbers satisfy circular test.

LET US SUM UP

Index number is a statistical measure or device with a purpose of showing average change in one or more related variables over time and space.

— Price index numbers are more commonly used. It measures relative changes in prices over a time period.

— We can have either simple or weighted index number. Simple index is also called unweighted index number or index number with equal weights. In this unit, we have discussed about the simple index numbers as well as weighted index number.

— We have discussed four properties of relatives: Identity property, time reversal property, circular property and modified circular property.

— We have learnt about two methods of constructing simple index numbers: Simple aggregate method and Method of simple average of relatives. The formula to find index number using simple aggregate method is given by

Again, the formula to find price index number using the method of simple averages of relatives is given by .

— We have also learnt about weighted index numbers. There are two groups of methods to calculate index number of this category: (a) Weighted aggregate method and (b) Weighted average of price relative methods. We briefly discussed about three types of weighted aggregate methods. These methods are popularly known as (1) Laspeyre’s Method, (2) Paasche’s Method and (3) Fisher’s Method.

— We have also learnt about the test of adequacy of index numbers. We mentioned three properties that a good index number should have. Fisher’s index number satisfies both the time reversal and factor reversal tests and hence it is known as Fisher’s Ideal Index Number.