#### Chapter 18 :-

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.

From the data given below construct the index number for the year 2021 taking 2020 as base year.

We can solve the above problem as given below.

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.

We can solve the above problem as given below.

\(\mathbf{P_{01}={{{\frac{ΣP_{1}}{ΣP_{0}}} }} × 100} \)
\(\mathbf{={{{\frac{2060}{1760}} }} × 100} \) = 117.05
Construct an index number for 2020 taking 2010 as base (Simple aggregative price method).

Solution is given below.

\(\mathbf{P_{01}={{{\frac{ΣP_{1}}{ΣP_{0}}} }} × 100} \)
\(\mathbf{={{{\frac{300}{250}} }} × 100} \) = 120

Construct index numbers of price from the following data by applying Laspeyre’s method.

Let us solve the above question as given below.

Laspeyre’s Method:
\(\mathbf{P_{01}={{{\frac{ΣP_{1}Q_{0}}{ΣP_{0}Q_{0}}} }} × 100} \)
Where,
ΣP

Using the data given below data, construct price index by Paasche’s method.

Let us solve the above question as given below.

\(\mathbf{P_{01}={\frac{ΣP_{1}Q_{1}}{{ΣP_{0}Q_{1}}} × 100}} \)
\(\mathbf{P_{01}={\frac{159}{{127}} × 100}} \) = 125.2
From the following data, construct price index by using

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.

From the following data construct an index for 2020 taking 2019 as base by the average of price relatives method.

Let us solve the above question as given below.

\({{\mathbf{P_{01} = {\frac{1}{{n}}}}}}\)Σ\({{\mathbf{\frac{P_{1}}{{P_{0}}} × 100}}}\)
\({{\mathbf{ = {\frac{616.6}{{5}}}}}}\)
= 122.32

**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 18.5 | ||||||
---|---|---|---|---|---|---|

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**II. Method of Averaging Relatives**

- (i) Simple average price relative index
- (ii) Weighted average price relative index

**(i) Simple Average Price Relative Index**

- P
_{1}is the price of commodity in the current period - P
_{1}is the price of commodity in the base period - \({{\frac{P_{1}}{{P_{0}}} × 100}} \) is the price relative
- n stands for the number of commodities

Table 18.13 | ||||||
---|---|---|---|---|---|---|

Commodity | Price in 2019 | Price 2020 | ||||

A | 50 | 70 | ||||

B | 40 | 60 | ||||

C | 80 | 90 | ||||

D | 110 | 120 | ||||

E | 20 | 20 |

Table 18.14 | ||||||
---|---|---|---|---|---|---|

Commodity | Price in 2019 P |
Price 2020 P |
Price Relative \({{\frac{P_{1}}{{P_{0}}} × 100}} \) |
|||

A | 50 | 70 | 140.0 | |||

B | 40 | 60 | 150.0 | |||

C | 80 | 90 | 112.5 | |||

D | 110 | 120 | 109.1 | |||

E | 20 | 20 | 100.0 | |||

Σ\({{\mathbf{\frac{P_{1}}{{P_{0}}} × 100}}}\) = 611.6 |

**(ii) Weighted Average Price Relative Index**