Unit 2: Index Number-II
2.1 Tests for Index Numbers
To be considered a good ("ideal") index, a formula should satisfy certain mathematical properties. These tests check the logical consistency of the index formula.
Notation:
P₀₁ = Price index for period 1 (current) based on period 0 (base).P₁₀ = Price index for period 0 (current) based on period 1 (base).Q₀₁ = Quantity index for period 1 based on period 0.V₀₁ = Value index for period 1 based on period 0 = Σp₁q₁ / Σp₀q₀
2.2 Time Reversal Test
The Test: An index formula satisfies this test if, when the base period and current period are interchanged, the two resulting index numbers are reciprocals of each other.
P₀₁ * P₁₀ = 1
Logic: If prices doubled from 2020 to 2024 (P₀₁ = 2), then from 2024's perspective, 2020 prices should be half (P₁₀ = 1/2). Their product should be 1.
Checking the Formulas:
- Laspeyre's (L):
- P₀₁(L) = Σp₁q₀ / Σp₀q₀
- P₁₀(L) = Σp₀q₁ / Σp₁q₁ (Swap 0 and 1, but 'q' is always base, which is now '1')
- P₀₁(L) * P₁₀(L) = (Σp₁q₀ / Σp₀q₀) * (Σp₀q₁ / Σp₁q₁) ≠ 1
- Conclusion: Laspeyre's FAILS the test.
- Paasche's (P):
- P₀₁(P) = Σp₁q₁ / Σp₀q₁
- P₁₀(P) = Σp₀q₀ / Σp₁q₀ (Swap 0 and 1, 'q' is always current, which is now '0')
- P₀₁(P) * P₁₀(P) = (Σp₁q₁ / Σp₀q₁) * (Σp₀q₀ / Σp₁q₀) ≠ 1
- Conclusion: Paasche's FAILS the test.
- Fisher's (F):
- P₀₁(F) = sqrt[ (Σp₁q₀/Σp₀q₀) * (Σp₁q₁/Σp₀q₁) ]
- P₁₀(F) = sqrt[ (Σp₀q₁/Σp₁q₁) * (Σp₀q₀/Σp₁q₀) ]
- P₀₁(F) * P₁₀(F) = sqrt[ (A/B)*(C/D) * (D/C)*(B/A) ] = sqrt(1) = 1
- Conclusion: Fisher's SATISFIES the test. (This is one reason it's "Ideal".)
2.3 Factor Reversal Test
The Test: An index formula satisfies this test if, when the price (p) and quantity (q) "factors" are interchanged, the product of the resulting Price Index and Quantity Index equals the true Value Index.
P₀₁ * Q₀₁ = V₀₁ = Σp₁q₁ / Σp₀q₀
Logic: The total change in Value (Price × Quantity) should be the product of the change in Price and the change in Quantity.
Checking the Formulas:
- Laspeyre's (L):
- P₀₁(L) = Σp₁q₀ / Σp₀q₀
- Q₀₁(L) = Σq₁p₀ / Σq₀p₀ (Swap 'p' and 'q' in the formula)
- P₀₁(L) * Q₀₁(L) = (Σp₁q₀ / Σp₀q₀) * (Σq₁p₀ / Σq₀p₀) ≠ V₀₁
- Conclusion: Laspeyre's FAILS the test.
- Paasche's (P):
- P₀₁(P) = Σp₁q₁ / Σp₀q₁
- Q₀₁(P) = Σq₁p₁ / Σq₀p₁ (Swap 'p' and 'q' in the formula)
- P₀₁(P) * Q₀₁(P) = (Σp₁q₁ / Σp₀q₁) * (Σq₁p₁ / Σq₀p₁) ≠ V₀₁
- Conclusion: Paasche's FAILS the test.
- Fisher's (F):
- P₀₁(F) = sqrt[ (Σp₁q₀/Σp₀q₀) * (Σp₁q₁/Σp₀q₁) ]
- Q₀₁(F) = sqrt[ (Σq₁p₀/Σq₀p₀) * (Σq₁p₁/Σq₀p₁) ] (Swap 'p' and 'q')
- P₀₁(F) * Q₀₁(F) = sqrt[ (Σp₁q₀/Σp₀q₀) * (Σp₁q₁/Σp₀q₁) * (Σq₁p₀/Σq₀p₀) * (Σq₁p₁/Σq₀p₁) ]
- P₀₁(F) * Q₀₁(F) = sqrt[ (Σp₁q₁/Σp₀q₀) * (Σp₁q₁/Σp₀q₀) ] (Terms cancel out)
- P₀₁(F) * Q₀₁(F) = Σp₁q₁ / Σp₀q₀ = V₀₁
- Conclusion: Fisher's SATISFIES the test. (This is the other reason it's "Ideal".)
2.4 Chain Index Numbers
The methods we've seen so far are Fixed Base Index Numbers (e.g., 2010=100, 2011=105, 2012=112... all compared to 2010).
Chain Index Numbers are different. Each period is compared to the immediately preceding period. This creates a "chain."
Process:
- Step 1: Calculate Link Relatives. This is an index for each period with the previous period as the base.
- Link Relative for Period 2 = (P₂ / P₁) * 100
- Link Relative for Period 3 = (P₃ / P₂) * 100
- Step 2: Chain the Links. Multiply the current link relative by the previous chain index (and divide by 100).
- Chain Index for Period 1 = 100
- Chain Index for Period 2 = (Link Relative for 2 * Chain Index for 1) / 100
- Chain Index for Period 3 = (Link Relative for 3 * Chain Index for 2) / 100
Pros and Cons:
- Pros:
- Allows for adding or removing items from the "basket" easily, as the weights can be updated yearly.
- Uses a more recent, relevant base for comparison.
- Cons:
- More complex to calculate.
- May have "drift" or "error" that accumulates over long periods.
2.5 Consumer Price Index (CPI) Numbers
Definition
The Consumer Price Index (CPI), also known as the Cost of Living Index, is the most well-known index. It measures the average change over time in the prices paid by urban consumers for a fixed basket of consumer goods and services (food, housing, transport, medical care, etc.).
It is the primary measure of inflation.
Construction
Constructing the CPI is a massive, complex task:
- Family Budget Surveys: A survey is done on thousands of families to determine what they buy and how much of their budget they spend on each item (e.g., 25% on food, 30% on housing...). This determines the "basket" and, more importantly, the weights (w).
- Price Collection: The government collects tens of thousands of price quotes for all the items in the basket from stores all over the country.
- Calculation: The CPI is calculated using a weighted index formula. Most countries use a variation of Laspeyre's method (a fixed basket/weights).
CPI = [ Σ(p₁w) / Σ(p₀w) ] * 100
This is also known as the Weighted Aggregative Expenditure Method, which is identical to Laspeyre's formula (w = p₀q₀).
Applications
- Measure of Inflation: The percentage change in the CPI is the inflation rate.
- Wage/Pension Adjustment: Used to calculate "cost-of-living adjustments" (like Dearness Allowance) to protect salaries from losing purchasing power.
- Economic Policy: The central bank (like the RBI) uses the CPI to make decisions about interest rates to control inflation.
- Deflating: Converting nominal wages to "real wages" to see if purchasing power has actually increased.
Limitations
- Substitution Bias: As a Laspeyre's-type index, it has an upward bias because it assumes a fixed basket (it ignores the fact that people switch to chicken if beef prices rise).
- New Goods Bias: The "basket" is slow to update. It doesn't account for the value of new products (e.g., smartphones, streaming services) when they first appear.
- Quality Change Bias: It struggles to account for improvements in product quality (a 2024 car is better than a 2010 car).
- Not for Everyone: The "average" basket for an "urban consumer" may not reflect the spending patterns of a rural family or a wealthy individual.