Fisher’s ideal Method (TRT & FRT)

Fisher combined the best of both above-mentioned formulas which resulted in an ideal method. This method uses both current and base year quantities as weights as follows:

P =  √[ (∑P₁Q₀÷∑P₀Q₀) × (∑P₁Q₁÷∑P₀Q₁) ]  ×100

NOTE: Index number of base year is generally assumed to be 100 if not given

Fisher’s Method is an Ideal Measure

As noted Fisher’s method uses views of both Laspeyres and Paasche. Hence it takes into account the prices and quantities of both years. Moreover, it is based on the concept of the geometric mean, which is considered as the best mean method.

However, the most important evidence for the above affirmation is that it satisfies both time reversal and factor reversal tests. Time reversal test checks that when we reverse the current year to base year and vice-versa, the product of indexes should be equal to unity. This confirms the working of a formula in both directions. Also, factor reversal test implies that interchanging the piece and quantities do not give varying results. This proves the consistency of the formula.

Common Problems with Construction of Index Numbers

Due to the availability of a wide range of index numbers we have to select an index number that matches the objective we want to fulfill. For example, to study the impact of a change in the government’s budget on people, one should refer to the price index number.

It must be noted that the selected base year should be a normal one. In other words, there should be no reforms in that year which can influence the economy in a drastic manner. If such is chosen as the base year there will be a big variation in the index numbers, which would not reflect the accurate changes over the years.

Also, it is not possible to include all the goods and services along with their prices in our calculations. This means we need to select various goods and services that can effectively represent all of them. In a word, a sample size has to be selected. Larger the sample size more is the accuracy. And we need to select the method of calculation that suits best with the objective in hand.

Tests of consistency;

1. TRT: Time reversal test

P01 * P10 = 1

TRT is not satisfied by Laspeyre’s price index and Paache’s price index, but it’s satisfied by Fisher’s price index.

2. FRT: Factor reversal test

P01 * Q01 = V01

FRT is satisfied only by Fisher’s price index.

We can notice that Fisher’s price index satisfies both time reversal and factor reversal test. This is one of the reason why Fisher’s price index is known as the ideal index number. The other reason is that this index considers both the current and base year quantities.

Unit Test

This test states that the formula for constructing an index number should be independent of the units in which prices and quantities are expressed. All methods, except simple aggregative method, satisfy this test.

Circular Test:

According to this, if indices are constructed for year one based on year zero, for year two based on year one and for year zero based on year two, the product of all the indices should be equal to 1.

Symbolically:

P01 X P12 X P20 = 1

This test is satisfied by

• Simple aggregative method and
• Kelly’s method.