Index Number Adequacy

Many formulae have been suggested for constructing an index number and the problem is that of selecting the most appropriate one in a given case. The following tests are suggested for choosing an appropriate index:

Unit test: - The unit test requires that the formula for constructing an index must be independent of the units in which the prices and quantities are quoted, except for the simple (unweighted) aggregative index.

Time reversal test: - proof. Irving fisher has made a careful study of the different proposals for computing an index number and has suggested satisfactory. The two most important of these calls are the time reversal test and the factor reversal test.

The Time reversal test is a test to determine whether a given technique will work both ways in time forwards and backward. In the words of fisher, The test is the formula for calculating the index number and it must be such that it will give the same ratio between one point of comparison and the other, no matter which of the time is taken by a base, in other words when the data for any two years are treated by the same technique, but with the bases reversed the two index numbers secured must be reciprocals of each other so that their product is unity symbolically the following relation must be satisfied.

p₀₁ x p₁₀ = 1

Where p₀₁ is the index for time 1 on time 0 as base and p₁₀ the index for time 0 on time 1 as base. If the product is not unity there is said to a time bias in the technique. Thus if from 2008 to 2009 the price of rice increased from $ 480. To $560 per quintal the price in 2006 must be 1333/1/3 percent of the price in 2008 and the price in 2008 must be 75 percent of the price in 2009. One figure is the reciprocal of the other their product (1.33 x 0.75) is unity this is obviously true for each individual price relative and according to the reversal test, it must be true for the index number.

   Factor reversal test:- another test suggested by fisher is commonly known as factor reversal test. It holds that the product of a price index and the quantity index must be usual to the corresponding values index. In another words of fisher. Just as each formula should allow the interchange of the two times without giving inconsistent results so it ought to allow the interchange of the prices and quantities without giving inconsistent result the two results multiply altogether must give the true value ratio. In other words the test is that the change in price multiplied by the change in quantity must be equal to the total change in value. The total value of a given commodity in a given year and the product of the quantity and the price per unit (total value = p x q). The ratio of the total value in 1 year to the total value in the preceding year I sp1q1 / p0 q0 if from one year to the next. Both price and quantity could double the price relative would be 200, the quantity relative 200, and the value relative 400.  The totals value in the present year and the base year respectively and if p01 shows the change in price in the present year and Q01 the change in quantity in the present year. Then

p₀₁ x q₀₁ = Σp₁ q₁ / Σ p₀ q₀

If the product is not equal to the value ratio, there is with reference to this test. An error in one or both forte index numbers.

The factor reversal test is satisfied only by the fisher ideal index

Proof: p₀₁ = √(Σ p₁q₀/Σp₀q₀) X Σ p₁ q₁ / Σ p₀ q₁

Changing p to q and q to p

q₀₁ √(Σ q₁p₀/Σq₀p₀) X Σ q₁ p1 / Σ qo p1

p₀₁ X q₀₁ = √(Σ p₁q₀/Σp₀q₀ )  X Σp₁q₁/Σp₀q₁  X Σ q₁ p₀/Σq₀p₀  X Σq₁p₁/Σq₀p₁

= √((Σ p₁q₁)²/(Σp₀q₀)²) = Σ p₁q₁ / Σ p₀ q₀