Index Numbers
An index number is a statistical device that measures relative changes in a variable or a group of variables over time with reference to a base period. Index numbers are often called the "barometers of economic activity."
Features of Index Numbers
- They are expressed as ratios or percentages (base period = 100).
- They measure relative change, not absolute change.
- They are used to compare over time, place, or conditions.
Uses of Index Numbers
- 1.Measuring changes in price levels (inflation)
- 2.Measuring cost of living
- 3.Measuring industrial and agricultural production
- 4.Used as deflators to convert nominal to real values
- 5.Useful in formulating economic policy
Types of Index Numbers
1. Price Index Numbers: Measure changes in prices. E.g., Wholesale Price Index (WPI), Consumer Price Index (CPI).
2. Quantity Index Numbers: Measure changes in volume of production or consumption.
3. Value Index Numbers: Combine price and quantity changes.
Simple (Unweighted) Index Numbers
Simple Aggregative Method:
Price Index = (Sum of prices in current year / Sum of prices in base year) × 100
P01 = (Sum(P1) / Sum(P0)) × 100
Simple Average of Price Relatives Method:
First compute price relative for each item: P = (P1/P0) × 100
Then Price Index = Sum(P) / N
Weighted Index Numbers
Laspeyre's Index (Uses base year quantities as weights):
P01 = (Sum(P1 × Q0) / Sum(P0 × Q0)) × 100
Paasche's Index (Uses current year quantities as weights):
P01 = (Sum(P1 × Q1) / Sum(P0 × Q1)) × 100
Fisher's Ideal Index (Geometric Mean of Laspeyre's and Paasche's):
P01 = √(Laspeyre's × Paasche's)
Fisher's index is considered "ideal" because it satisfies both the time reversal test and the factor reversal test.
Consumer Price Index (CPI)
CPI measures the change in the cost of a fixed basket of goods and services for consumers. It is used to measure inflation and adjust wages, pensions, and contracts.
Real Wage = (Nominal Wage / CPI) × 100
This converts money wages into purchasing power terms.
Inflation Rate
Inflation Rate = ((CPI(current year) - CPI(previous year)) / CPI(previous year)) × 100
Worked Examples
Base year price of rice = Rs 20, current year = Rs 30. Compute price relative.
Price Relative = (30 / 20) × 100 = 150 (prices rose by 50%)
Simple aggregative index: Base year prices: 10, 20, 30 (Sum = 60). Current year: 12, 25, 36 (Sum = 73).
P01 = (73 / 60) × 100 = 121.67
Laspeyre's Index: Two goods.
| Good | P0 | Q0 | P1 |
|------|----|----|-----|
| A | 5 | 10 | 8 |
| B | 4 | 15 | 6 |
Sum(P1×Q0) = 8×10 + 6×15 = 80 + 90 = 170
Sum(P0×Q0) = 5×10 + 4×15 = 50 + 60 = 110
Laspeyre's = (170/110) × 100 = 154.55
CPI in 2020 = 150. A worker earned Rs 12,000 per month. Find real wage.
Real Wage = (12,000 / 150) × 100 = Rs 8,000
If CPI rose from 120 to 132 in one year, find the inflation rate.
Inflation = ((132 - 120) / 120) × 100 = (12/120) × 100 = 10%
Fisher's Ideal Index: If Laspeyre's = 150 and Paasche's = 144.
Fisher's = √(150 × 144) = √(21600) = 146.97
Why is Fisher's index called "ideal"?
Because it is the geometric mean of Laspeyre's (base-year weighted) and Paasche's (current-year weighted) indices, balancing their biases. It satisfies both the time reversal test (P01 × P10 = 1) and the factor reversal test.
Common mistakes
Common mistakes
Confusing Laspeyre's (base-year quantities) with Paasche's (current-year quantities). Laspeyre's index tends to OVERESTIMATE inflation (old weights); Paasche's tends to UNDERESTIMATE it. Also remember: the base year index is always 100.
Summary
Index numbers measure relative changes using a base period value of 100. Key types include simple aggregative, simple average of relatives, Laspeyre's, Paasche's, and Fisher's indices. The CPI measures cost of living; real wages are obtained by deflating nominal wages by the CPI. Fisher's is "ideal" as it satisfies both reversal tests.