# Index Numbers

**Index number** is a specialized average designed to measure the
change in the level of an activity or item, either with respect to time or
geographic location or some other characteristic. It is described either
as a ratio or a percentage. For example, when we say that consumer price index
for 1998 is 175 compared to 1991, it means that consumer prices have risen by
75% over these seven years.

*Wholesale Price Index*(WPI) and *Consumer Price Index* (CPI) are
widely used terms. They indicate the inflation rates, and also changes in
standard of living. Consumer price index is based on prices of five sets of
items - Food, Housing (Rent), Household goods, Fuel and light, and
Miscellaneous. Each item is based on study of a number of items - e.g. Food
includes Rice, Wheat, Dal, Milk, and so on.

### Methods of constructing index numbers

**Price Relative** means the ratio of price of a certain item in current
year to the price of that item in base year, expressed as a percentage (i.e.
**Price Relative = (p _{2}/p_{1})×100**). For example, if a
colour TV cost Rs 12000 in 1981 and Rs. 18000 in 1998, the price relative is

(18000/12000)×100 = 150.

Generally instead of one item, rates of a number of items are given, for current year as well as for base year. Sometimes different weights, or quantities are also given for those items. There are a number of ways to calculate index numbers in such cases.

### Unweighted index numbers

**Simple Aggregative**

**Simple Average of Price Relatives**

### Weighted (or Arithmetic Mean method)

**Weighted Aggregative**

**Weighted Average of Price Relatives**

where I = (p_{2}/p_{1})×100

## Illustrative Examples

### Example

During a certain period, the cost of living index number goes from 110 to 200 and the salary of a worker is also raised from Rs 325 to Rs. 500. Does the worker really gains or loses, and by how much amount in real terms?

### Solution

Real wage = [(Actual wage)/(cost of living index)] × 100

So real wage of Rs 325 = (325/110) × 100 = Rs 295·45

and real wage of Rs 500 = (500/200) × 100 = Rs 250

So the worker actually loses, i.e. Rs (295·45 -250)

= Rs 45·45 in real terms.

### Example

With price index of 1991 as 100, the cost of living index for 1996 is 160 and for 1997 is 180. The salary of an employee increased from Rs 5000 in 1996 to Rs 5500 in 1997. Find out whether the real income in 1997 has increased or decreased as compared to 1996. Also calculate if any extra dearness allowance should be paid to him to compensate for loss. Also calculate purchasing power of rupee in 1996 and 1997.

### Solution

The price relative of 1997 compared to 1996 is

I = (p_{2}/p_{1})×100 = (180/160)×100 = 112·50.

Hence the wages of Rs 5500 in 1997 are equal to wages of

(5500×100)/112.50 = Rs 4889 in 1996.

Thus we find that real income in 1997 has decreased compared to 1996.

Now let us assume that Rs x extra dearness allowance is paid to compensate for
this loss.

Hence (5500+x)/180 = 5000/160 => x = Rs 125

Now **purchasing power** of a rupee is defined as

Purchasing power = (Index number for base year)/(Index number for current year)

So purchasing power of rupee in 1996 compared to 1991 is

Rs 100/160 = paise (100/160) × 100 = 62·5 paise
62 paise

Purchasing power of rupee in 1997 compared to 1991 is

Rs = 100/180 paise = (100 × 100)/180 = 55·55 paise
56 paise

Thus we see that due to inflation (price rise), the purchasing power of rupee is falling.

### Example

Using 1985 as base year, the index numbers for the price of a commodity in 1986 and 1987 are 118 and 125. Calculate the index numbers for 1985 and 1987 if 1986 is taken as the base year.

### Solution

Let p_{1}, p_{2}, p_{3} be prices in 1985, 1986,
1987.

Then p_{1/}p_{2} × 100 = 118 and p_{3/}p_{2} ;× 100 = 125

Now price index of 1985 with 1986 as base

= p_{1/}p_{2} × 100 = (100/118) × 100 = 84·75

Price index of 1987 with 1986 as base

=

This procedure is called **shifting of base**.

## Exercise

- Find by simple aggregate method, the index number of the following data:

**Commodity****Base Price****Current Price**Rice 140 180 Oil 400 550 Sugar 100 250 wheat 125 150 Fish 200 300 - Construct index number for above data using price relatives.
- Calculate a cost of living index from the following table of prices and weights.

Weight Price index Food 35 108.5 Rent 9 102.6 Clothes 10 97.0 Fuel 7 100.9 Miscellaneous 39 103.7 - Construct Index number for following data:

Butter Bread Tea Bacon Relative Index 181 116 110 152 Weight 4 12 3 7 - Construct the consumer price index number for 1998 on the basis of 1988
from the following data, using simple aggregates.

Commodity A B C D E 1988 Price per unit(Rs) 16.00 40.00 0.50 5.12 2.00 1988 Price per unit(Rs) 20.00 60.00 0.50 6.25 1.50 - Redo above problem using price relatives.
- Based on year 1988 as base, the index numbers for 1988, 1989, 1990, 1991 and 1992 are 100, 110,120, 200 and 400. Now taking 1992 as base year, calculate index numbers for years 1988, 1989,1990, 1991 and 1992.
- Taking 1995 as the base year, with an index number 100, calculate an
index number for 1999, based on (i) simple aggregate (ii) price relatives
derived from the table given below:

Commodity A B C D Price per unit in 1995 20 10 25 40 Price per unit in 1999 24 20 30 40 - Taking 1995 as the base year with an index number 100, calculate an index
number for 1995 based on the weighted average of price relatives derived from
the table given below:

Commodity A B C D Price per unit in 1995 10 20 5 40 Price per unit in 1995 30 35 10 80 Weight 20 30 10 40 - Net monthly income of an employee was Rs 3000 per month in 1990 and Rs 5000 per month in 1998. Consumer price index in 1990 was 150. Find the consumer price index for 1998, given that net income of the employee is linked to consumer price index (so that his standard of living does not decrease).

## Answers

**1.**148·19

**2.**157·21

**3.**Index ==104.4

**4.**Index = = 135

**5.**Consumer price index = 138·71

**6.**Consumer price index = 114·41

**7.**25·0, 27·5, 30·0, 50·0, 100

**8.**(i) 114·00 (ii) 135·00

**9.**Index = = 212·5

**10.**250