Methods of Constructing Index Numbers
by Index numbers can be constructed using different statistical methods depending on the purpose of the study, the availability of data, and the importance assigned to different commodities. The main methods are classified into Simple (Unweighted) Methods and Weighted Methods.
(A) Simple (Unweighted) Methods
1. Simple Aggregative Method
The simple aggregative method is one of the easiest methods of constructing index numbers. Under this method, the prices of all selected commodities in the current year are added together, and their total is divided by the sum of prices in the base year. The result is multiplied by 100 to obtain the price index.
Formula: Price Index = (ΣP₁ / ΣP₀) × 100
Where:
P₁ = Current-year prices
P₀ = Base-year prices
2. Simple Average of Price Relatives Method
Under this method, price relatives are calculated for each commodity by dividing its current-year price by its base-year price and multiplying the result by 100. The average of these price relatives gives the index number. The arithmetic mean or geometric mean may be used to calculate the average.
Formula using Arithmetic Mean:
Price Index = ΣR / N
Where:
R = Price relative of each commodity
N = Number of commodities
Price Relative = (P₁ / P₀) × 100
(B) Weighted Methods
1. Weighted Aggregative Method
The weighted aggregative method assigns weights to commodities according to their relative importance. Weights may represent quantities consumed, produced, or sold. This method is generally more representative than simple methods because it recognises that commodities do not have equal importance.
2. Laspeyres’ Method
Laspeyres’ method uses base-year quantities as weights to calculate the price index. It compares current-year prices with base-year prices while keeping the quantities constant at the base-year level.
Formula: Laspeyres’ Price Index = (ΣP₁Q₀ / ΣP₀Q₀) × 100
Where:
Q₀ = Base-year quantities
This method is relatively easy to calculate because base-year quantity data can be used throughout the comparison.
3. Paasche’s Method
Paasche’s method uses current-year quantities as weights. It measures the change in prices by comparing the cost of current-year quantities at current prices with their cost at base-year prices.
Formula: Paasche’s Price Index = (ΣP₁Q₁ / ΣP₀Q₁) × 100
Where:
Q₁ = Current-year quantities
This method reflects current consumption or purchasing patterns but requires updated quantity data for each comparison period.
4. Fisher’s Ideal Method
Fisher’s Ideal Method combines Laspeyres’ and Paasche’s price indices by calculating their geometric mean. It considers both base-year and current-year quantities, making it a balanced approach to measuring price changes.
Formula: Fisher’s Ideal Price Index = √(Laspeyres’ Index × Paasche’s Index)
This method is called ideal because it satisfies important statistical tests, including the time reversal test and factor reversal test, under the standard index-number framework.
5. Marshall–Edgeworth Method
The Marshall–Edgeworth method uses the sum of base-year and current-year quantities as weights. It considers quantity information from both periods and therefore avoids relying exclusively on either base-year or current-year quantities.
Formula: Marshall–Edgeworth Price Index = [ΣP₁(Q₀ + Q₁) / ΣP₀(Q₀ + Q₁)] × 100
This method can provide a balanced comparison when quantity data for both periods are available.
6. Dorbish–Bowley Method
The Dorbish–Bowley method calculates the arithmetic mean of Laspeyres’ and Paasche’s price indices. It combines the two indices to provide a measure that considers both base-year and current-year quantity weights.
Formula: Dorbish–Bowley Price Index = (Laspeyres’ Index + Paasche’s Index) / 2
This method is relatively simple to understand and calculate. However, unlike Fisher’s Ideal Method, it does not generally satisfy both the time reversal and factor reversal tests.