index number

Introduction

In this lesson we shall deal with index numbers which are specialised averages.

An index number is a specialised measure designed to show changes in a variable or a group of related variables with respect to time, geographical location or other characteristics. — Spiegal

Mr. Chandran was abroad for his studies. He came back to Kerala after four years having completed his studies. One day Chandran went to the nearby market. He found that prices of most commodities had changed. Some items have become costlier, while others cheaper. He told his mother about the price changes and it was bewildering to both. Similarly, the prices of various industrial out- puts are changing — increasing or decreasing.

Look at the following cases:

  1. An agricultural labourer in Kerala was getting ₹ 50 per day in 1980. Today he gets ₹ 500 per day. Does it mean that his standard of living has risen 10 times ? By how much should his salary be raised so that he is better off as before ?
  2. You might have read in business newspapers, statements such ‘Sensex’ crossing 30,000 mark and a single day rise’ of 800 points made to increase wealth of investors by 165,352 crores What exactly is SENSEX ?
  3. Within a matter of a few months, price of petroleum products has gone up by 25%. Government says, inflation rate will go up due to rise in petroleum products. How does one measure inflation ?
These are a few of many questions you confront in your daily life. Hence a single measure like average will not suffice to depict the characteristics of these related variables. So, there must be a handy statistical tool to compare changes in a group of related variables. Index number is such a measure.

What is an Index Number ?

An index number is a statistical device used to measure changes in the magnitude of a group of related variables.

Index numbers are usually expressed in terms of percentages. It measures the effect of changes over a period of time. There are two periods — current period and the base period. The period with which comparison is made is known as base period. The period for which comparison is made is known as current period. The value in the base period is given the index number 100. If you want to know how much the price changed in 2017 for the level in 2000, then 2000 becomes the base. The index number of any period is in proportion with it. Thus an index number of 300 indicates that the value is three times that of the base period.

Characteristics:

  1. Index numbers are specialised averages.
  2. Index numbers measure the net change in a group of related variables.
  3. Index numbers measure the effect of changes over a period of time.

Uses of Index Numbers:

  1. They help in framing suitable policies.
  2. They reveal trends and tendencies.
  3. Index numbers are very useful in deflating.
  4. They help in measuring the purchasing power of money.

Construction of Index Numbers

There are two methods of constructing an index number. They are:

  • I. Aggregative method, and
  • II. Method of averaging relatives.

I. Aggregative method

There are two types of index numbers under this category. They are: (i) simple aggregative price index and (ii) Weighted aggregative price index.

(i) Simple Aggregative Price Index

The formula for a simple aggregative price index is:

index number

Where,

P01 = Index number of the current year

ΣP1 = Total of current year prices of all commodities

ΣP0 = Total of base year prices of all commodities

Steps

  • i) Add the current year prices of all commodities to get ΣP1
  • ii) Add the base year prices of all commodities to get ΣP0
  • iii) Divide ΣP1 by ΣP0 and multiply the quotient by 100
  • From the data given below construct the index number for the year 2021 taking 2020 as base year.

  • Table 18.1
    Unit Commodities Price (in ₹)
    2020 2021
    Wheat quintal 200 250
    Rice " 300 400
    Pulses " 400 500
    Milk litre 2 3
    Clothing meter 3 5

    We can solve the above problem as given below.

    Table 18.2
    Unit Commodities Price (in ₹)
    2020 (P0) 2021 (P1)
    Wheat quintal 200 250
    Rice " 300 400
    Pulses " 400 500
    Milk litre 2 3
    Clothing meter 3 5
    ΣP0 = 905 ΣP1 = 1158

    Index Number of 2021 or

    \(\mathbf{P_{01}={{{\frac{ΣP_{1}}{ΣP_{0}}} }} × 100} \)

    \(\mathbf{={{{\frac{1158}{905}} }} × 100} \) = 127.96

    It means that the prices in 2021 were 27.96% higher than the prices of 2020.

  • From the following data construct price index by simple aggregative method.
  • Table 18.3
    Commodity A B C D E F
    Price in 2000 210 310 100 240 420 480
    Price 2016 260 300 160 340 460 540

    We can solve the above problem as given below.

    Table 18.4
    Commodity Price in 2000 P0 Price 2016 P1
    A 210 260
    B 310 300
    C 100 160
    D 240 340
    E 420 460
    F 480 540
    ΣP0 = 1760 ΣP1 = 2060

    \(\mathbf{P_{01}={{{\frac{ΣP_{1}}{ΣP_{0}}} }} × 100} \)

    \(\mathbf{={{{\frac{2060}{1760}} }} × 100} \) = 117.05

  • Construct an index number for 2020 taking 2010 as base (Simple aggregative price method).
  • Table 18.4
    Commodity Price in 2010 Price 2020
    A 210 260
    B 310 300
    C 100 160
    D 240 340

    Solution is given below.

    Table 18.6
    Commodity Price in 2010 P0 Price 2020 P1
    A 210 260
    B 310 300
    C 100 160
    D 240 340
    ΣP0 = 250 ΣP1 = 300

    \(\mathbf{P_{01}={{{\frac{ΣP_{1}}{ΣP_{0}}} }} × 100} \)

    \(\mathbf{={{{\frac{300}{250}} }} × 100} \) = 120

    (ii) Weighted Aggregative Price Index

    The index numbers discussed above assign equal importance to all the items included in the index. Construction of useful index numbers requires assigning of weight to each commodity according to its importance in the total phenomenon. This will make the index number more representative.

    There are different methods of assigning weights. A weighted aggregative price index using base period quantities as weights is known as Laspeyre’s price index. A weighted aggregative price index using current period quantities as weights is known as Paasche’s price index.

    Laspeyre’s Price Index

    The formula is,

    Laspeyre’s Price Index

    Steps

    • (i) Multiply the current year price of each commodity with base year quantity to get P1 Q0, and then find ΣP0.
    • (ii) Multiply the base year price of each commodity with the base year quantity to get p0 Q0 and then find ΣP0Q0.
    • (iii) Apply the formula, \(\mathbf{P_{01}{{{\frac{ΣP_{1}Q_{0}}{ΣP_{0}Q_{0}}} }} × 100} \)

  • Construct index numbers of price from the following data by applying Laspeyre’s method.
  • Table 18.7
    Commodity 2018 2019
    Price Quantity Price Quantity
    A 2 8 4 5
    B 5 10 6 9
    C 4 14 5 13
    D 2 19 2 10

    Let us solve the above question as given below.

    Table 18.8
    Commodity 2018 2019 P1Q0 P0Q0
    Price P0 Quantity Q0 Price P1 Quantity Q0
    A 2 8 4 5 5 5
    B 5 10 6 9 5 5
    C 4 14 5 13 5 5
    D 2 19 2 10 5 5
    ΣP1Q0 = 200 ΣP0Q0 = 160

    Laspeyre’s Method:

    \(\mathbf{P_{01}={{{\frac{ΣP_{1}Q_{0}}{ΣP_{0}Q_{0}}} }} × 100} \)

    Where,

    ΣP1Q0 = 200,

    ΣP0Q0 = 160

    \({P_{01}={{{\frac{200}{160} }}} × 100} \) = 125

    Paasche’s Price Index

    The formula is,

    Paasche’s Price Index

    Steps

    • (i) Multiply the current year price of each commodity with current year quantity to get P1 Q0 and then find ΣP1 Q1.
    • (ii) Multiply the base year price of each commodity with current year quantity to get P0 Q1 and then find ΣP0 Q1.
    • (iii) Apply the formula, \(\mathbf{P_{01}={\frac{ΣP_{1}Q_{1}}{{ΣP_{0}Q_{1}}} × 100}} \)

  • Using the data given below data, construct price index by Paasche’s method.
  • Table 18.9
    Commodity 2018 2019
    Price Quantity Price Quantity
    A 2 8 4 5
    B 5 10 6 9
    C 4 14 5 13
    D 2 19 2 10

    Let us solve the above question as given below.

    Table 18.10
    Commodity 2018 2019 P1Q1 P0Q1
    Price P0 Price P1 Quantity Q1
    A 2 4 5 20 10
    B 5 6 9 54 45
    C 4 5 13 65 52
    D 2 2 10 20 20
    ΣP1Q1 = 159 ΣP0Q1 = 127

    \(\mathbf{P_{01}={\frac{ΣP_{1}Q_{1}}{{ΣP_{0}Q_{1}}} × 100}} \)

    \(\mathbf{P_{01}={\frac{159}{{127}} × 100}} \) = 125.2

  • From the following data, construct price index by using
    • (i) Laspeyre’s Method
    • (ii) Paasche’s Method

    Table 18.11
    Commodity 2018 2019
    Price Amount Paid Price Amount Paid
    A 6 90 15 150
    B 9 54 12 84
    C 4 100 10 300
    D 3 21 8 80
    E 4 40 7 56

    Here we have to arrive the quantity from the price and amount paid. Quantity is obtained by dividing amount paid by the price of each commodity.

    Table 18.12
    Commodity P0 q0 P1 q1 P0q0 P1q0 P0q1 P1q1
    A 6 15 15 10 90 225 60 150
    B 9 6 12 7 54 72 63 84
    C 4 25 10 30 100 250 120 300
    D 3 7 8 10 21 56 30 80
    E 4 10 7 8 40 70 32 56
    ΣP0q0 = 305 ΣP1q0 = 673 ΣP0q1 = 305 ΣP1q1 = 670

    (i) Laspeyre's Method

    \(\mathbf{P_{01}={\frac{ΣP_{1}Q_{0}}{{ΣP_{0}Q_{0}}} × 100}} \)

    \(\mathbf{P_{01}={\frac{673}{{305}} × 100}} \) = 220.66

    (ii) Paasche's Method

    \(\mathbf{P_{01}={\frac{ΣP_{1}Q_{1}}{{ΣP_{0}Q_{1}}} × 100}} \)

    \(\mathbf{P_{01}={\frac{670}{{305}} × 100}} \) = 219.67