Unit 1: Index Number-I
1.1 Definition of Index Numbers
An index number is a specialized statistical measure that shows the change in a variable (or group of related variables) over time, or between different locations or categories.
It's a way to express a complex change in a single, simple number. The most common type is a price index, which measures the change in the general price level of a group of items.
- Base Period: The period against which all comparisons are made. The index for the base period is always 100.
- Current Period: The period for which we want to measure the change.
If the price index for 2024 is 115 (with 2020 as the base year), it means that, on average, prices have increased by 15% from 2020 to 2024.
1.2 Problems in the Construction of Index Numbers
Creating a good index number is difficult. Here are the main challenges:
- Purpose of the Index: What are we measuring? An index for consumer prices (like food) will be different from one for industrial production. The purpose must be clear.
- Selection of the Base Period: The base year should be a "normal" or "stable" year, free from unusual events like wars, famines, or economic crises. A bad base year will distort all comparisons.
- Selection of Items (The "Basket"): We can't include every single item. We must select a representative sample. (e.g., for a food price index, we'd include rice and milk, but maybe not expensive imported cheese).
- Obtaining Price/Quantity Data: It can be hard to get accurate, representative prices from various locations.
- Choice of Average: Should we use the Arithmetic Mean, Geometric Mean, or Median? The Geometric Mean is often preferred as it's less affected by outliers.
- Choice of Weights: This is the most important problem. Should all items be treated equally (unweighted), or should more important items (like rice) have a greater impact (weight) on the index than less important items (like salt)?
1.3 Unweighted Index Numbers
These methods assume all items in the basket are equally important. They are simple but often unrealistic.
Notation:
p₀ = Price of an item in the Base periodq₀ = Quantity of an item in the Base periodp₁ = Price of an item in the Current periodq₁ = Quantity of an item in the Current periodP₀₁ = Price Index of the Current period (1) based on the Base period (0)
1. Simple Aggregative Method
Adds up all current prices and divides by the sum of all base prices.
P₀₁ = ( Σp₁ / Σp₀ ) * 100
Major Flaw: This method is heavily distorted by items with high prices (e.g., a car, even if rarely bought) and by the units of measurement (e.g., measuring rice in grams vs. kilograms will completely change the index).
2. Simple Average of Price Relatives Method
A "price relative" is (p₁/p₀) for a single item. This method averages these relatives.
P₀₁ = ( 1/n ) * Σ( p₁ / p₀ ) * 100
This is better than the aggregative method as it is not affected by units of measurement.
1.4 Weighted Index Numbers
These methods are more realistic. They assign a weight (w) to each item based on its relative importance (e.g., how much money is spent on it). The "weight" is usually the value (Price × Quantity).
Weighted Aggregative Methods
This is the most common category. The general formula is:
P₀₁ = ( Σ(p₁w) / Σ(p₀w) ) * 100
The difference between methods (Laspeyre's, Paasche's, etc.) is simply in what they choose for the weight 'w'.
1.5 Laspeyre's Index
- Weight Used: Base period quantities (q₀).
- Logic: It measures the change in cost of buying the same basket of goods from the base year at current prices.
P₀₁ (L) = ( Σ(p₁q₀) / Σ(p₀q₀) ) * 100
Pros: Easy to calculate (base quantities `q₀` don't change each year). Easy to compare over time.
Cons: Tends to overestimate inflation. It assumes people keep buying the same (base year) basket, even when prices change (in reality, people substitute cheaper goods).
1.6 Paasche's Index
- Weight Used: Current period quantities (q₁).
- Logic: It measures the change in cost of buying today's basket of goods at current prices versus base-year prices.
P₀₁ (P) = ( Σ(p₁q₁) / Σ(p₀q₁) ) * 100
Pros: Reflects current consumption patterns.
Cons: Tends to underestimate inflation (due to substitution bias). Very difficult to calculate (you need new quantity data every period). Hard to compare year-to-year.
1.7 Marshall-Edgeworth Index
- Weight Used: The average of the base and current period quantities. w = (q₀ + q₁) / 2
- Logic: A compromise between Laspeyre's and Paasche's.
P₀₁ (ME) = [ Σ(p₁ * (q₀ + q₁)) / Σ(p₀ * (q₀ + q₁)) ] * 100
This can be rewritten as:
P₀₁ (ME) = [ (Σp₁q₀ + Σp₁q₁) / (Σp₀q₀ + Σp₀q₁) ] * 100
1.8 Fisher's Ideal Index
- Logic: Another compromise, considered the "Ideal" index. It is the Geometric Mean of Laspeyre's and Paasche's indexes.
P₀₁ (F) = sqrt( P₀₁(L) * P₀₁(P) )
Expanding this:
P₀₁ (F) = sqrt( [ (Σp₁q₀ / Σp₀q₀) ] * [ (Σp₁q₁ / Σp₀q₁) ] ) * 100
Why is it "Ideal"?
- It balances the upward bias of Laspeyre's and the downward bias of Paasche's.
- It satisfies important statistical tests (Time Reversal and Factor Reversal tests), which we will see in the next unit.
1.9 Applications and Limitations of Index Numbers
Applications
- Economic Barometers: They are used to measure the "health" of an economy (e.g., Consumer Price Index (CPI), Stock Market Indices like Sensex).
- Inflation/Deflation: The CPI is the official measure of inflation.
- Wage Adjustments: Used to adjust salaries and pensions (e.g., Dearness Allowance) to maintain purchasing power.
- Policy Making: Government uses indices to formulate economic, fiscal, and monetary policies.
- Deflating: Used to convert "nominal" values (at current prices) to "real" values (at constant, base-year prices) to see true growth.
Real Value = (Nominal Value / Price Index) * 100
Limitations
- Only Approximations: They are based on samples, so they are not perfectly accurate.
- Can be Misleading: A poor choice of base year, weights, or items can create a biased index.
- Ignores Quality Changes: A phone in 2024 is much more expensive than one in 2010, but it's also a vastly superior product. An index number struggles to capture this change in quality.
- Tastes and Habits Change: The "basket of goods" from 10 years ago might be irrelevant today, making long-term comparisons difficult.