# Construction of an Index Number

Index numbers are not constructed simply for measuring changes in the value of money and cost of living. They can be employed for measuring any quantitative changes say in wages, imports, exports, employment and industrial activity. There are many difficulties in the construction of index numbers. It is very difficult to assist weight to each commodity according to its importance in national consumption. Wheat assumes more importance in the laboring class budgets than in those of rich persons. Cable TV in urban India has higher significance than in rural India. Even in rural areas different people assign different weights to TV. The same difficulty arises in the case of other commodities which we use, such as two wheelers or cars.

Another problem that arises here is the fact that people change their buying practices from year to year. Changes in forecasting and taste may change consumption habits. Many commodities which were regarded as luxuries in the early 1900's are considered to be necessities now. So, the index number constructed for measuring the changes in the value of money does not give us a correct indication of its changes. Another difficulty which is associated with the construction of index number is the choice of a base year. The base year should be a year of normal economic activity.

There are two methods of constructing an index number. It can be computed by the aggregative method and by the method of averaging relatives.

# The Aggregative Method

The formula for a simple aggregative price index is

Example: Calculate the simple aggregative price index on the basis of following data
 Commodity Price (Rs.) in 1990 Price (Rs.) in 2000 Rice 120 180 Wheat 80 100 Oil 300 400 Pulses 130 180 Sugar 150 200

Solution:
 The total price of commodities in the two years is given by:

Using the formula the simple aggregative price index is given by :

The formula for a weighted aggregative price index is:

Example: Calculate the weighted aggregative price index number on the basis of the following data:
 Commodity Base Year Price (Rs.) Current Year Price (Rs.) Weights (W) A 150 200 5 B 300 320 6 C 180 250 7 D 75 150 10 E 40 50 2

Solution: Based on the data given above we calculate the following:
 Commodity A B C D E 750 1800 1260 750 80 1000 1920 1750 1500 100

Using the formula the weighted aggregative price index number is obtained as :

The weights taken are either from the base period or from the current period (but the choice of period remains the same throughout ) based on the discretion of the statistician.

An index number becomes a weighted index when the relative importance of items is taken care of. Here weights are quantity weights. To construct a weighted ggregative index, a well specified basket of commodities is taken and its worth each year is calculated. It thus measures the changing value of a fixed aggregate of goods. Since the total value changes with a fixed basket, the change is due to price change. Various methods of calculating a weighted aggregative index use different baskets with respect to time.

The two most basic formulas used to calculate price indices are the Paasche index and the Laspeyres index. The Laspeyres and Paasche price indices are both measures of the overall price level. Both are calculated using the cost of a set of commodities during the base period and the current period. If we choose the weights in the base, period we have the Laspeyres price index and if we choose the weights in the current period, we have the Paasche price index.

Example: Calculate Laspeyre's and Paasche's price index numbers on the basis of the following data:
 Commodity A B C D E Base Year Price 10 25 30 15 20 Current Year Price 15 40 45 30 25 Base Year Quantity 6 10 15 20 8 Current Year Quantity 8 20 12 15 6

Solution : Based on the data given the following calculations are done:

 Commodity A B C D E 10 25 30 15 20 6 10 15 20 8 15 40 45 30 25 8 20 12 15 6 60 250 450 300 160 90 400 675 600 200 80 500 360 225 120 120 800 540 450 150

We get the required Indices to be:
Laspeyre's Price Index:

# Method of Averaging relatives

For a single commodity, the price index is the ratio of its price in the current period to that in the base period, normally expressed in terms of percentage. The method of averaging relatives takes the average of these relatives for all the commodities in question. The price index number using price relatives is defined as

where and represent the price of the commodity in the current period and base period respectively. The ratio is also called the price relative of the commodity and stands for the number of commodities.

The weighted index of price relatives is the weighted arithmetic mean of price relatives defined as

where Wi is the weight of the ith commodity.

In a weighted price relative index, weights may be determined by the fraction of the total expenditure spent on them during the base period or the current period depending on the formula that is to be used.

The best-known index number is the consumer price index, which measures changes in retail prices paid by consumers. In addition, a cost-of-living index (COLI) is a price index number that measures relative cost of living over time.