Factor Reversal Test

Another test suggested by Fisher is known as factor reversal test. It holds that the product of a price index and the quantity index should be equal to the corresponding value index. In the words of Fisher, “Just as each formula should permit the interchange the prices and quantities without giving inconsistent results, so it ought to permit the interchange of the two times without giving inconsistent results, i.e., the two results multiplied together should give the true value ratio.” In other words, the test is that the change in price multiplied by the change in quantity should be equal to the total change in value. The total value of a given commodity in a given year is the product of the quantity and the price per unit (total value = p x q). The ratio of the total value in one year to the total value in the preceding year is p1q1/p0q0. If from one year to the next, both price and quantity should double, the price relative would be 200, the quantity relative 200, and the value relative 400. The total value in the second year would be four times the value in the first year. In other words, if p1 and p0 represent price and q1 and q0 the quantities in the current year and the base year, respectively, and if p01 represents the change in price in the current year and Q01 the change in quantity in the current year, then


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

The factor reversal test is satisfied only by the Fisher’s Ideal Index.

Proof:


Change p to q and q to p




Since the factor reversal test is satisfied by the Fisher’s Ideal index. This means, of course, that the formula serves equally well for constructing indices of quantities as for constructing indices of prices, the quantity index being derived by interchanging p and q in the ideal formula.

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