Understanding Price Index Numbers and the Factor Reversal Test

Price index numbers are statistical tools used to measure the average change in the prices of a basket of goods and services over time or between different places. They are essential for understanding concepts like inflation and changes in purchasing power. Different methods exist for calculating price index numbers, each with its own formula and properties.

One important property that statisticians use to evaluate the quality of an index number is called the Factor Reversal Test.

What is the Factor Reversal Test?

The Factor Reversal Test, proposed by Irving Fisher, checks whether an index number formula holds true when the roles of price and quantity are interchanged. Basically, it states that the product of a price index (P) and the corresponding quantity index (Q), calculated using the same formula, should be equal to the value index (V).

The value index is simply the ratio of the total value of the goods and services in the current period to the total value in the base period. Value is calculated as price multiplied by quantity.

Mathematically, the Factor Reversal Test is satisfied if:

\( P_{01} \times Q_{01} = V_{01} \)

Where:

• \( P_{01} \) is the price index for period 1 relative to period 0.
• \( Q_{01} \) is the quantity index for period 1 relative to period 0.
• \( V_{01} \) is the value index for period 1 relative to period 0.

The value index is calculated as:

\( V_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} \)

Where:

• \( p_0 \) and \( q_0 \) are price and quantity in the base period (period 0).
• \( p_1 \) and \( q_1 \) are price and quantity in the current period (period 1).

Examining Common Price Index Formulas

Let's look at some common price index formulas and see if they satisfy the Factor Reversal Test.

Which Price Index Satisfies the Factor Reversal Test?

Let's test the given options:

1. Laspeyres Price Index:

• Laspeyres price index (\( P_{01}^L \)) uses base period quantities as weights.
• Laspeyres quantity index (\( Q_{01}^L \)) uses base period prices as weights.
• \( P_{01}^L \times Q_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum q_1 p_0}{\sum q_0 p_0} \)
• This product is generally not equal to \( \frac{\sum p_1 q_1}{\sum p_0 q_0} \). So, Laspeyres index does not satisfy the Factor Reversal Test.

2. Paasche's Price Index:

• Paasche's price index (\( P_{01}^P \)) uses current period quantities as weights.
• Paasche's quantity index (\( Q_{01}^P \)) uses current period prices as weights.
• \( P_{01}^P \times Q_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum q_1 p_1}{\sum q_0 p_1} \)
• This product is also generally not equal to \( \frac{\sum p_1 q_1}{\sum p_0 q_0} \). So, Paasche's index does not satisfy the Factor Reversal Test.

3. Walsh's Price Index:

• Walsh's index uses a weighted average of quantities from both periods (specifically, the geometric mean of the base and current period quantities) as weights.
• While Walsh's index has desirable properties like satisfying the Time Reversal Test, it does not satisfy the Factor Reversal Test.

4. Fisher's Price Index:

• Fisher's Price Index (\( P_{01}^F \)), also known as Fisher's Ideal Index, is the geometric mean of the Laspeyres and Paasche price indices.
• Fisher's Quantity Index (\( Q_{01}^F \)) is the geometric mean of the Laspeyres and Paasche quantity indices.
• Let's check if it satisfies the test:
• \( P_{01}^F \times Q_{01}^F = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times \sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1}} \)
• \( = \sqrt{\frac{\sum p_1 q_0 \times \sum p_1 q_1}{\sum p_0 q_0 \times \sum p_0 q_1} \times \frac{\sum q_1 p_0 \times \sum q_1 p_1}{\sum q_0 p_0 \times \sum q_0 p_1}} \)
• Rearranging the terms under the square root:
• \( = \sqrt{\left(\frac{\sum p_1 q_1}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_1 q_0}{\sum p_0 q_1}\right) \times \left(\frac{\sum q_1 p_0}{\sum q_0 p_1}\right) \times \left(\frac{\sum q_1 p_1}{\sum q_0 p_0}\right)} \)
• Looking closely, we can pair terms:
• \( = \sqrt{\left(\frac{\sum p_1 q_1}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_1 q_0}{\sum p_0 q_1}\right) \times \left(\frac{\sum p_0 q_1}{\sum p_1 q_0}\right) \times \left(\frac{\sum p_1 q_1}{\sum p_0 q_0}\right)} \) (Note: \( \sum q_1 p_0 = \sum p_0 q_1 \) and \( \sum q_1 p_1 = \sum p_1 q_1 \))
• \( = \sqrt{\left(\frac{\sum p_1 q_1}{\sum p_0 q_0}\right)^2 \times \left(\frac{\sum p_1 q_0}{\sum p_0 q_1} \times \frac{\sum p_0 q_1}{\sum p_1 q_0}\right)} \)
• \( = \sqrt{\left(\frac{\sum p_1 q_1}{\sum p_0 q_0}\right)^2 \times 1} \)
• \( = \frac{\sum p_1 q_1}{\sum p_0 q_0} \)
• This result is exactly the value index \( V_{01} \). Thus, Fisher's Ideal Index satisfies the Factor Reversal Test.

Therefore, among the given options, Fisher's price index is the one that satisfies the Factor Reversal Test.

Revision Table: Price Index Tests

Additional Information on Index Numbers

Index numbers are widely used in economics and statistics. They help track changes in various economic indicators like prices, quantities, industrial production, etc. Choosing the right index number formula depends on the specific purpose of the study and the properties desired.

Besides the Factor Reversal Test, another important test is the Time Reversal Test. This test requires that the index number calculated backward in time (from period 1 to period 0) should be the reciprocal of the index number calculated forward in time (from period 0 to period 1). Mathematically, this is \( P_{01} \times P_{10} = 1 \).

Fisher's Ideal Index is called "Ideal" because it satisfies both the Time Reversal Test and the Factor Reversal Test, which are considered important consistency criteria for index numbers.

Other types of indices include the Marshall-Edgeworth index and the Tornqvist index, which also have different properties regarding these tests.