## Karnataka 2nd PUC Statistics Notes Chapter 2 Index Numbers

**High Lights of the Topic:**

→ Index numbers were originally developed to show the effect of changes in prices on the cost of living. But now is developed to study Index Number in the field of Whole sale prices, Retail prices, Industrial production, Agricultural production, Exports, Imports and in all stages of Planning, Policy making. And so Index numbers are called as Economic Barometers.

→ “Index number is the ratio, which gives the average change in the level of phenomenon between two different periods of time or places”.

OR

→ According Spiegel “An Index Number is a statistical measure designed to show an average change in a variable or group of related variables with respect to Time, Geographical location or Income etc.”

For example,

1. If index number of wholesale price of certain commodities in 2012 as compares to 2005 is 125, this implies that overall level of wholesale prices of commodities in 2012 has increased by 25% of the level in 2005.

2. Index number for 2004 with base 1990 is 150, which means average price level has increased by 5 0% from 1990 to 2004.

3. Price index number is 125. Which means price level In the current year is 125% of the price in the base year.

4. The average price level of a commodity in the year 200.0 is 2\(\frac{1}{4}\) times of what it was in 1995. Which means Index number is 100 × 2\(\frac{1}{4}\) = 225.

→ In all above examples 2012, 2004 and 2000 are called current years and 2005, 1990, 1995 are called base years.

→ The year selected for comparison is called the base year’ and ‘the year for which comparison are required is called the current year’.

→ ‘The index for the base year is always taken to be 100’.

→ Prices and quantities in the base year are denoted by suffix ‘0’ as p_{0}, q\frac{1}{4} and that of current year are denoted by suffix 1 as p_{1}, q_{1}.

→ If price of a commodity is Rs.15/- in 2008 and Rs.21/- in 2013, the index number of price for the year 2013 with respect to the base 2008 is 140. That is, the price of the commodity in 2013 is 40% of the price in 2005. Here, only a single variable is considered the index number is called ‘Relative’. In this particular case, it is ‘price relative’.

→ ‘Price relative is the price in the current year (p_{1}) expressed as the percentage of the price in the base year (p_{1})’, and is given by: P=\(\frac{\mathrm{p}_{1}}{\mathrm{P}_{0}}\) × 100

→ If quantity is compared over a period, it is the quantity relative.

→ “Quantity relative is the ratio of quantity of a commodity in the current year (q_{1}) expressed as the percentage of the quantity in the base year (q_{0})”, and is given by: Q = \(\frac{q_{1}}{q_{0}}\) × 100.

→ If the value of a commodity is compared over a period, it is the value raltive.

→ “Value relative is the value of a commodity in the current year (v_{1}) expressed as the percentage of the value in the base year (v_{01})” and is given by: V_{01} = \(\frac{p_{1} q_{1}}{p_{0} q_{0}}\) × 100.

**Uses:**

The following are the main uses of index numbers:

→ Index numbers are useful to governments in formulation of decisions and policies regarding taxation, imports, exports, grant of licenses to new firms and in fixation of bank rates, salary, and grant of dearness allowances to employees.

→ To measure the trends and tendencies, comparison of variation in production, price, demand, supply, maintain stock, marketing the goods etc.,

→ Also Index Numbers are used in evaluation of purchasing power of money.

→ Index numbers simplifies the data thus facilitates comparative study.

→ Index numbers measures the change in cost of living (i.e., consumer price index number) of different groups of people over a period.

**Limitations:**

→ Many formulae are used in the construction and gives different values for the Index

→ As the customs and habits of the people changes from time to time, the uses of Items/ commodities also vary.

→ There is an ample scope for bias in the construction of Index number ie. Index numbers can be misused, to get desired conclusions

→ In construction of Index number only a few representative items are used, and so it does not indicate the overall changes.

→ While constructing Index numbers, the quality of product is not taken into consideration.

**Types of Index Numbers:**

- Price Index number.
- Quantity Index number
- Value Index number.

1. Price Index Number: Price index number indicates the general level of prices of articles in the current year as compared to that of base period, denoted by P_{01} – price index number current year ‘ 1’ to base year ‘0’

→ Price index numbers may be, the retail price index number, wholesale price index number and cost of living index number or consumer price index number are all price index numbers.

→ The wholesale price index number which treasures the relative change in the wholesale price of commodities.

→ The retail price index number which measures the relative change in the retail price of the commodities.

2. Quantity Index number: Quantity index numbers are the index numbers of quantity of . goods manufactured by a firm, exported or imported, quantity of agricultural produce etc.,

they indicate the physical output in an economy over a period of time. Denoted by Q0)

3. Value Index number: Value Index numbers study the relative change in the total money value (price multiplied by quantity) of production. Value index number indicates the effect of combination of price and quantity changes (transaction) in between two time periods. Denoted

**Construction of Index Number:**

The following are the steps/stages/principles/Heads involved:-

- Purpose and the scope
- Selection of base period
- Selection of commodities or items
- Selection of price list
- Selection an average
- Selection of weights
- Selection of formula

1. Purpose and scope: At the very outset the purpose for which the index numbers is being constructed should be clearly defined. Since most of the latter problems will depend upon purpose, because the selection of items, base year, formulae will be different for different purposes. The purpose may be such as cost of living index number for urban areas, rural areas, factory workers etc. should be clearly defined.

Also, it is necessary to state the scope of the index numbers such as, geographical region, time, Income etc.

2. Selection of the Base year: Index numbers are always constructed with reference to some period called base period. The period may be a year, a month, week or a day.

Following points are to be considered while selecting the base year.

(i) The base period selected should be Normal one, that is economically stable. It should be free from wars, floods, famine, booms, depressions, which affects the economic activities.

(ii) Also the base year selected should not be too long or too short distance. Since economic activities are always dynamic.

(iii) While selecting the base year, decision has to make whether fixed base or chain base year to be selected. In fixed base year, if the period of comparison is kept fixed for all current years, it is called fixed-base period. Where as in chain base year, comparison is made always with the previous year.

3. Selection of items: Items means sample of representative commodities/ items, required for day to day living, which are included in the construction of index number. According to the tastes, habits, customs and traditions etc., of working class people should be selected.It is also necessary to decide the grade or quality of items.

4. Selection of price list: After selecting the commodities/ items, the next problem is to fix the prices for the items selected. Since prices vary from place to place, and even from shop to shop in the same market. Hence the suitable way is to obtain the prices ascertained by Marketing inspectors, Super Bazaars, Co-operative Societies or took out the average prices collected by various shops or places where concerned.

5. Selection of average: Since index numbers are specialised averages s.o, in the construction of index numbers, suitable average should be used. The choice is made between Arithmetic mean and Geometric mean. Usually Arithmetic mean with weighted averages is popularly applied, but when accuracy is required Geometric mean is suitable.

6. Selection of Weights: Weight of the items means, the relative importance of an items used in the construction of index number. Since all’ items are not equally important in day-to-day usage. Proper weights should be attached to the various items depending on their relative importance, eg: Sugar-Rice, Atta-Salt etc. Mostly these weights are quantities in the base period or those in the current period are considered. Sometimes a combination of quantities of different time periods may be used as weights.

7. Selection of formula: A large number of formulae have been devised for construction of Index numbers. Hence a reasonable formula/ method suitable to the object and the data available should be selected.

**Methods of Constructing Index Numbers:**

Index numbers are constructed with the following methods

(A) Simple/Un weighted Index numbers

(B) Weighted Index numbers

Under each method index numbers are based on price relatives and price aggregative.

(A) Simple/Un weighted Index numbers: If the Index Numbers are constructed on the basis of prices of the items and not considering the respective weights [q] are called un weighted Index Numbers.

1. Simple Average of price relatives:

(i) Simple Arithmetic mean of price relatives

P_{01} = \(\frac{\Sigma P}{n}\); where P = \(\frac{p_{1}}{p_{0}}\) × 100; n = number of commodities in the group.

(ii) Simple Gepmetric mean of price realatives: p_{01} = A.L\(\left[\frac{\Sigma \log P}{n}\right]\)

(B) Weighted index numbers:- If the Index Numbers are constructed on the basis of prices of the- items and the respective weights [q] are called Weighted Index Numbers.

1. Weighted average of price relatives: In these Index Numbers value weights/relative importance [w] of the items are considered.

(i) Weighted AM. of price relatives: P_{01} = \(\frac{\sum \mathrm{P} w}{\sum w}\); where P = \(\frac{p_{1}}{p_{0}}\) × 100: w = weights.

(ii) Weighted GM of price relatives: P_{01} = A.L\(\left[\frac{\sum w \log \mathrm{P}}{\sum w}\right]\)

2. Weighted aggregate Price Index Numbers: In the weighted aggregative price index numbers, quantity weights are assigned to various items and the weighted aggregate of the prices are obtained.

(i) LASPEYRE’S price index: In this method the base period quantities (q_{0}) are taken as weights. p_{o1} = \(\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}\) × 100

(ii) PAASCHE’S price index: In this method the current year quantities (q_{1}) are taken as weights. P_{01} = \(\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}}\) × 100

(iii) MARSHALL-EDGE WORTH’S price index:- In this method the arithmetic mean of base year (q_{0}) and current year quantities (q_{1}) are taken as weights.

P_{01} = \(\frac{\sum p_{1}\left(\frac{q_{0}+q_{1}}{2}\right)}{\sum p_{0}\left(\frac{q_{0}+q_{1}}{2}\right)}\) × 100

After simplification we get: OR p_{01} = \(\frac{\sum p_{1} q_{0}+\sum p_{1} q_{1}}{\sum p_{0} q_{0}+\sum p_{0} q_{1}}\) × 100

(iv) DORBISH BOWLEY’S price index:

Here ‘arithmetic mean of has Laspeyre’s and Paasche’s price indices are considered in the formula’.

D – B = \(\frac{\mathrm{L}+\mathrm{P}}{2}\) is the realation between them;

p_{01} = \(\frac{1}{2}\left[\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}+\frac{\sum p_{1} q_{1}}{\sum p_{1} q_{0}}\right]\) × 100

(v) FISHER’S price index:

Here ‘geometric means of Laspeyre’s and paasche’s Indices are considered as the method’.

F = \(\sqrt{\mathrm{L} \times \mathrm{P}}\) is the relation between them.

P_{01} = \(\sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}+\frac{\sum p_{1} q_{1}}{\sum_{1} p_{0} q_{1}}} \times 100\) × 100

(iv) KELLY’S price index number

In this method specific weights to be fixed for all the periods. It is given by P_{01} = \(\frac{\sum p_{1} q}{\sum p_{0} q}\) × 100 Where q denotes the fixed quantity (weight) for both base and current period.

**Quantity Index Numbers:**

The Quantity index numbers study the changes in the volume of goods produced consumed, exported/imported in the current period as compared to base period. Prices are the weights in quantity index number.

Weighted Aggregative quantity Index numbers:

1. LASPEYRE’S Quantity index number:- In this method the base year prices (p_{0}) are taken as weights.

Q_{01} = \(\frac{\sum q_{1} p_{0}}{\sum q_{0} p_{0}}\) × 100

2. PAASCHE’S Quantity index number:- In this method the current year quantities (p_{0}) are taken as weights.

Q_{01} = \(\frac{\sum q_{1} \mathrm{P}_{1}}{\sum q_{0} \mathrm{P}_{1}}\) × 100

3. MARSHALL-EDGEWORTH’S Quantity index number:- In this method the arithmetic mean of base year (p_{0}) and current year quantities (p,) are taken as weights.

Q_{01} = \(\frac{\sum q_{1} p_{0}+\sum q_{1} p_{1}}{\sum q_{0} p_{1}+\sum q_{0} p_{1}}\) × 100

4. DORBISH BOWLEY’S Quantity index number: – Here ‘arithmetic mean of has Laspeyre’s and Paasche’s quantity indices are considered in the formula’ ie D – B = \(\frac{\mathrm{L}+\mathrm{P}}{2}\)

Q_{01} = \(\frac{1}{2}\left[\frac{\sum q_{1} p_{0}}{\sum q_{0} p_{0}}+\frac{\sum q_{1} p_{1}}{\sum q_{0} p_{1}}\right]\) × 100

5. FISHER’S Quantity index number: – Here ‘geometric means of Laspeyre’s and paasche’s Indices are considered as the method’, ie. F = \(\sqrt{L \times P}\)

Q_{01} \(\left[\frac{\sum q_{1} p_{0}}{\sum q_{0} p_{0}} \times \frac{\sum q_{1} p_{1}}{\sum q_{0} p_{1}}\right]\) × 100

**Tests For Adequacy of Index Numbers:**

The following tests are used to determine the suitability, consistency and reliability of the Index Numbers. They are:

- Unit Test
- Time Reversal Test
- Factor Reversal Test
- Circular Test

1. Unit test:

→ This test requires that the index number used for the construction of index number should be independent of the units (statistical units).

→ This test satisfies by all index number formulae except the simple aggregative of prices index formula.

2. Time Reversal Test: (TRT):

This test requires that the index number computed backwards should be the reciprocal of the index number computed forward, except the factor 100.

i. e. P_{o1} = \(\frac{1}{\mathrm{P}_{10}}\) OR P_{01} × P_{10} = 1 (Except the factor 100)

This test satisfies by Marshall-Edgeworth’s, Fisher’s and Kelly’s index numbers

3. Factor Reversal Test: (FRT):

This test requires that the product of the Price Index number and the Quantity Index number should be equal to the net change in the value taking place between two periods, except the factor 100.

i.e., P_{01} × Q_{01} = \(\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{0}}\) = V_{01} (Expect the factor 100)

This test satisfies only by Fisher’s

4. Circular test: This requires that the index number is to work in a circular manner and this property enables us to find the index numbers from period to period without referring back to the original base each time. For 3 years 0(1998), 1(2005) and 2 (2010):

Circular test is said to be satisfied if: P_{01} × P_{12} × P_{2o} = 1 (Except the factor 100)

Suppose when the test is verified to the simple aggregative method for the three years 0, 1, 2:

P_{01} × P_{12} × P_{2o} = \(\frac{\sum p_{1}}{\sum p_{0}} \times \frac{\sum p_{2}}{\sum p_{1}} \times \frac{\sum p_{0}}{\sum p_{2}}\) × 1

This test is satisfied only by simple geometric mean of the. price relatives and Kelly’s index number.

**Fisher’s method is known as ideal for following reasons:**

- It satisfies both time reversal tests and factor reversal tests
- It is free from bias in use of weights, ie, it takes both current and base year quantities as weights.
- It is based on geometric mean which is considered as the best average.

**Consumer Price Index Numbers**

{Cost of living index number}

→ The general price index number fails to give an idea of the effect of the change in the general price level on the cost of living of different class of people. Also the nature and the consumption of commodities are not same for all class of people.

→ Hence the other method which is suitable to calculate index number for a particular class of people is consumer price index number, also called as the cost of living index number is used “Cost of living index number is the index number of the cost met by a specified class of consumers in buying a basket of goods and services.

→ Here the basket of goods and services means various items and services needed in daily life of the specified consumers. Such as food, clothing, fuel and lighting, house rent etc. Here the class consumers mean a group of consumers having almost identical pattern of consumption. Generally those of factory workers, government employees, and the consumers who belonging to a particular community etc.

**Uses:**

Consumer price index numbers are used by governments to adjust from time to time- the wage policy, price policy, rent control, taxation, and capacity of retail prices, also in finding real rupee value etc. Specifically

- They are used in fixation of the salary and grant of Dearness Allowances to Government employees.
- They are used in evaluation of purchasing power of money and deflating the money.
- They are used for comparing the cost of living of different classes of people.

Construction of consumer price/ cost of living Index number:

The following are the main steps/stages/principles/points/Heads involved in the construction of cost of living index numbers:

- Object and scope.
- Conducting family budget survey
- Obtaining the price quotation.
- Method of construction.

1. Object and Scope: Decide the class of consumers for which index number is required. Such as whether index number is meant for government employees, bank employees, merchants, farmers, etc. also the geographical location, as that of locality, city or town, or a community of people who are having the similar consumption pattern should be decided.

2. Conducting family budget survey: A sample survey regarding the average expenses of families on various items of consumption is conducted. In the survey, the information regarding commodities consumed by the families, their quality and the respective budget is collected. The items included are under the heads of

- Food
- Clothing
- Fuel and lighting and
- Miscellaneous.

Each of these can be divided into smaller groups, as food may be including Wheat, Rice, Dhal, etc., which are consumed by the people for whom index is meant.

3. Obtaining Price Quotations: While constructing the cost of living index number, retail prices of commodities/ items are to be collected. The price lists are obtained from different agencies from different places. Then, they are averaged and these averages are used in the construction of index number. Here price lists of current period as well as base period should be collected.

4. Methods of construction: There are two methods of construction of Consumer price Index Number

1. Aggregative Expenditure Method (AEM): In this method base year quantities are

considered as weight, i.e. C.P.I = \(\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}\) × 100

2. Family Budget Method (FBM): In this method weighted A M of price relatives are used.

i.e. C.P.I = \(\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}\); where P = \(\frac{p_{1}}{p_{0}}\) × 100; W = weights (p_{0}q_{0}).