Chain Base Method
In this method, there is no fixed base period; the year immediately preceding the one for which the price index has to be calculated is assumed as the base year. Thus, for the year 1994 the base year would be 1993, for 1993 it would be 1992, for 1992 it would be 1991, and so on. In this way there is no fixed base and it keeps on changing.
The chief advantage of this method is that the price relatives of a year can be compared with the price levels of the immediately preceding year. Businesses mostly interested in comparing this time period rather than comparing rates related to the distant past will utilize this method.
Another advantage of the chain base method is that it is possible to include new items in an index number or to delete old times which are no longer important. This is not possible with the fixed base method. But the chain base method has the drawback that comparisons cannot be made over a long period.
In chain base,
Link relative of current years $${\text{ = }}\frac{{{\text{Price in the Current Year}}}}{{{\text{Price in the preceding Year}}}} \times 100$$
or
\[{P_{n – 1,n}} = \frac{{{P_n}}}{{{P_{n – 1}}}} \times 100\]
Example:
Find the index numbers for the following data taking 1980 as the base year.
Years

$$1974$$

$$1975$$

$$1976$$

$$1977$$

$$1978$$

$$1979$$

Price

$$18$$

$$21$$

$$25$$

$$23$$

$$28$$

$$30$$

Solution:
Year

Price

Link Relatives
$$ = \frac{{{P_n}}}{{{P_{n – 1}}}} \times 100$$ 
Chain Indices

$$1974$$

$$18$$

$$\frac{{18}}{{18}} \times 100 = 100$$

$$100$$

$$1975$$

$$21$$

$$\frac{{21}}{{18}} \times 100 = 116.67$$

$$\frac{{100 \times 116.67}}{{100}} = 116.67$$

$$1976$$

$$25$$

$$\frac{{25}}{{21}} \times 100 = 119.05$$

$$\frac{{116.67 \times 119.05}}{{100}} = 138.9$$

$$1977$$

$$23$$

$$\frac{{23}}{{25}} \times 100 = 92$$

$$\frac{{138.9 \times 92}}{{100}} = 127.79$$

$$1978$$

$$28$$

$$\frac{{28}}{{23}} \times 100 = 121.74$$

$$\frac{{127.79 \times 121.74}}{{100}} = 155.57$$

$$1979$$

$$30$$

$$\frac{{30}}{{28}} \times 100 = 107.14$$

$$\frac{{155.57 \times 107.17}}{{100}} = 166.68$$

Selection of a suitable average
There are different averages which can be used in averaging the price relatives or link relatives of different commodities. Experts have suggested that the geometric mean should be calculated to average these relatives. But as the calculation of the geometric mean is difficult, it is mostly avoided and the arithmetic mean is commonly used. In some cases the median is used to remove the effect of wild observations.
Selection of suitable weights
In the calculation of price index numbers all commodities are not of equal importance. In order to give them due importance, commodities are given due weights. Weights are of two kinds: (a) Implicit weights and (b) explicit weights.
Implicit weights are not explicitly assigned to any commodity, but the commodity to which greater importance is attached and is repeated a number of times. A number of varieties of such commodities are included in the index number as separate items. Thus, if an index number wheat is to receive a weight of 3 and rice a weight of 2, three varieties of wheat and two varieties of rice included in these method weights are not apparent, but items are implicitly weighted.
Explicit weights are explicitly assigned to commodities. Only one variety of the commodity is included in the construction of the index number but its price relative is multiplied by the figure of weight assigned to it. Explicit weights are decided on a logical basis. For example, if wheat and rice are to be weighted in accordance with the value of their net output and if the ratio of their net output is 5:2, wheat would receive a weight of five and rice of two.
Sometimes the quantities which are consumed are used as weights. These are called quantity weights. The amount spent on different commodities can also be used as their weights. These are called the value weights.