20
May

**Introduction**

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

**Look at the following cases:**

- 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 ?
- 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 ?
- 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 ?

**What is an Index Number ?**

**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:**

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

**Uses of Index Numbers:**

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

**Construction of Index Numbers**

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

**I. Aggregative method**

**(i) Simple Aggregative Price Index**

_{01}= Index number of the current year ΣP

_{1}= Total of current year prices of all commodities ΣP

_{0}= Total of base year prices of all commodities

**Steps**

- i) Add the current year prices of all commodities to get ΣP
_{1} - ii) Add the base year prices of all commodities to get ΣP
_{0} - iii) Divide ΣP
_{1}by ΣP_{0}and multiply the quotient by 100

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 |

Table 18.2 | ||||||
---|---|---|---|---|---|---|

Unit | Commodities | Price (in ₹) | ||||

2020 (P_{0}) |
2021 (P_{1}) |
|||||

Wheat | quintal | 200 | 250 | |||

Rice | " | 300 | 400 | |||

Pulses | " | 400 | 500 | |||

Milk | litre | 2 | 3 | |||

Clothing | meter | 3 | 5 | |||

ΣP_{0} = 905 |
ΣP_{1} = 1158 |

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 |

Table 18.4 | ||||||
---|---|---|---|---|---|---|

Commodity | Price in 2000 P_{0} |
Price 2016 P_{1} |
||||

A | 210 | 260 | ||||

B | 310 | 300 | ||||

C | 100 | 160 | ||||

D | 240 | 340 | ||||

E | 420 | 460 | ||||

F | 480 | 540 | ||||

ΣP_{0} = 1760 |
ΣP_{1} = 2060 |

Table 18.4 | ||||||
---|---|---|---|---|---|---|

Commodity | Price in 2010 | Price 2020 | ||||

A | 210 | 260 | ||||

B | 310 | 300 | ||||

C | 100 | 160 | ||||

D | 240 | 340 |

Table 18.6 | ||||||
---|---|---|---|---|---|---|

Commodity | Price in 2010 P_{0} |
Price 2020 P_{1} |
||||

A | 210 | 260 | ||||

B | 310 | 300 | ||||

C | 100 | 160 | ||||

D | 240 | 340 | ||||

ΣP_{0} = 250 |
ΣP_{1} = 300 |

**(ii) Weighted Aggregative Price Index**

**Laspeyre’s Price Index**

**Steps**

- (i) Multiply the current year price of each commodity with base year quantity to get P
_{1}Q_{0}, and then find ΣP_{0}. - (ii) Multiply the base year price of each commodity with the base year quantity to get p
_{0}Q_{0}and then find ΣP_{0}Q_{0}. - (iii) Apply the formula, \(\mathbf{P_{01}{{{\frac{ΣP_{1}Q_{0}}{ΣP_{0}Q_{0}}} }} × 100} \)

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 |

Table 18.8 | ||||||
---|---|---|---|---|---|---|

Commodity | 2018 | 2019 | P_{1}Q_{0} |
P_{0}Q_{0} |
||

Price P_{0} |
Quantity Q_{0} |
Price P_{1} |
Quantity Q_{0} |
|||

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 |

ΣP_{1}Q_{0} = 200 |
ΣP_{0}Q_{0} = 160 |

_{1}Q

_{0}= 200, ΣP

_{0}Q

_{0}= 160 \({P_{01}={{{\frac{200}{160} }}} × 100} \) = 125

**Paasche’s Price Index**

**Steps**

- (i) Multiply the current year price of each commodity with current year quantity to get P
_{1}Q_{0}and then find ΣP_{1}Q_{1}. - (ii) Multiply the base year price of each commodity with current year quantity to get P
_{0}Q_{1}and then find ΣP_{0}Q_{1}. - (iii) Apply the formula, \(\mathbf{P_{01}={\frac{ΣP_{1}Q_{1}}{{ΣP_{0}Q_{1}}} × 100}} \)

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 |

Table 18.10 | ||||||
---|---|---|---|---|---|---|

Commodity | 2018 | 2019 | P_{1}Q_{1} |
P_{0}Q_{1} |
||

Price P_{0} |
Price P_{1} |
Quantity Q_{1} |
||||

A | 2 | 4 | 5 | 20 | 10 | |

B | 5 | 6 | 9 | 54 | 45 | |

C | 4 | 5 | 13 | 65 | 52 | |

D | 2 | 2 | 10 | 20 | 20 | |

ΣP_{1}Q_{1} = 159 |
ΣP_{0}Q_{1} = 127 |

- (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 |

Table 18.12 | ||||||||
---|---|---|---|---|---|---|---|---|

Commodity | P_{0} |
q_{0} |
P_{1} |
q_{1} |
P_{0}q_{0} |
P_{1}q_{0} |
P_{0}q_{1} |
P_{1}q_{1} |

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 |

ΣP_{0}q_{0} = 305 |
ΣP_{1}q_{0} = 673 |
ΣP_{0}q_{1} = 305 |
ΣP_{1}q_{1} = 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