Index Numbers
Index numbers measure the relative change in a variable or group of variables over time, compared to a base period. They convert absolute values into relative numbers, making comparisons easy. Essential tools in business and economics for tracking prices, production, wages, and living standards.
Concept and Uses
An index number expresses a value as a percentage of a base period value. If rice cost Rs 50/kg in 2020 (base) and Rs 65/kg in 2024, the price index = (65/50) × 100 = 130, meaning a 30% increase. Uses: measuring inflation (CPI), tracking stock market (NEPSE index), comparing living standards, deflating nominal values to real values, and guiding wage/tariff policy.
Types
Price index: measures price changes (CPI, WPI). Quantity index: measures quantity changes (Index of Industrial Production). Value index: measures total value changes (price × quantity). Simple index: single item. Composite index: group of items (most practical index numbers).
Simple Index Numbers
Simple price relative: P = (Pₙ/P₀) × 100. Simple aggregate: index = (ΣPₙ/ΣP₀) × 100 — gives equal weight to all items (expensive items dominate). Simple average of relatives: index = ΣR/n — averages percentage changes equally.
Weighted Index Numbers
Laspeyres' Index uses base period quantities as weights: P₀₁(L) = (ΣPₙQ₀/ΣP₀Q₀) × 100. Advantage: constant weights, easy comparison. Disadvantage: overestimates price increase (ignores substitution). Paasche's Index uses current period quantities: P₀₁(P) = (ΣPₙQₙ/ΣP₀Qₙ) × 100. Advantage: reflects current consumption. Disadvantage: underestimates, requires new data each period. Generally: Laspeyres ≥ True Index ≥ Paasche.
Fisher's Ideal Index
Fisher's Ideal = √(Laspeyres × Paasche). Called 'ideal' because it satisfies Time Reversal Test (P₀₁ × P₁₀ = 1) and Factor Reversal Test (price index × quantity index = value index). Neither Laspeyres nor Paasche alone satisfies both.
Consumer Price Index (CPI)
Measures changes in cost of a representative basket of goods consumed by households. Primary measure of inflation. Nepal's CPI compiled by Central Bureau of Statistics (CBS). Components: food, housing, clothing, transport, education, health. Used to: calculate inflation rate, adjust wages (dearness allowance), index government payments, deflate GDP, guide NRB monetary policy.
Base Year and Splicing
Base year should be normal (no wars, disasters), recent, with reliable data. Splicing links two series with different bases. Chain-base method computes index relative to preceding period, then chains together.
Summary
Index numbers — simple, weighted, Laspeyres, Paasche, Fisher — measure relative changes over time. CPI measures inflation affecting wages, policy, and economic decisions. Understanding construction, properties, and limitations is essential for interpreting economic data.
Worked Example: Laspeyres, Paasche & Fisher Index
Calculate price index numbers for 2024 (current year) with 2020 as base year using the following data:
| Commodity | P₀ (2020) | Q₀ (2020) | Pₙ (2024) | Qₙ (2024) | PₙQ₀ | P₀Q₀ | PₙQₙ | P₀Qₙ |
|---|---|---|---|---|---|---|---|---|
| Rice (per kg) | 50 | 100 | 70 | 90 | 7000 | 5000 | 6300 | 4500 |
| Oil (per litre) | 180 | 20 | 220 | 18 | 4400 | 3600 | 3960 | 3240 |
| Sugar (per kg) | 80 | 30 | 100 | 25 | 3000 | 2400 | 2500 | 2000 |
| Dal (per kg) | 120 | 25 | 160 | 22 | 4000 | 3000 | 3520 | 2640 |
| Total | Σ=18400 | Σ=14000 | Σ=16280 | Σ=12380 |
Laspeyres’ Index = (ΣPₙQ₀ / ΣP₀Q₀) × 100 = (18400/14000) × 100 = 131.43
Paasche’s Index = (ΣPₙQₙ / ΣP₀Qₙ) × 100 = (16280/12380) × 100 = 131.50
Fisher’s Ideal Index = √(131.43 × 131.50) = √17293.05 = 131.47
Interpretation: Prices have increased by approximately 31.5% from 2020 to 2024. Laspeyres (131.43) slightly underestimates compared to Paasche (131.50) in this case because consumers reduced quantities of items with larger price increases (substitution effect). Fisher’s index (131.47) provides the best estimate as it satisfies both the time reversal and factor reversal tests.
Verification — Time Reversal Test for Fisher’s: P₀₁ × P₁₀ should equal 1 (or 100² = 10000 when using percentage form). This confirms Fisher’s index is “ideal.”
CPI and Inflation Calculation
Example: Nepal’s CPI was 180 in Shrawan 2080 and 195 in Shrawan 2081 (base year 2071/72 = 100). Calculate the inflation rate.
Solution: Inflation rate = [(CPI₁ − CPI₀) / CPI₀] × 100 = [(195 − 180) / 180] × 100 = (15/180) × 100 = 8.33%
Interpretation: The general price level increased by 8.33% over the year. This affects: wage negotiations (workers demand at least 8.33% raise to maintain purchasing power), business pricing (firms may raise prices), NRB monetary policy (may tighten to control inflation), and government budgeting (social spending needs to increase to maintain real value).
Real vs Nominal Values: If a worker’s nominal salary rose from Rs 30,000 to Rs 32,000 (6.67% increase) but inflation was 8.33%, their real salary actually declined: Real salary = (32000/195) × 100 = Rs 16,410 in base year terms vs (30000/180) × 100 = Rs 16,667 previously. Despite a nominal raise, the worker is worse off in real terms.
Exam Tips for Index Numbers
Tip 1: Set up a clear table with all required columns (P₀, Q₀, Pₙ, Qₙ, and products) — tables prevent calculation errors. Tip 2: Remember which formula uses which quantities: Laspeyres = base quantities (Q₀), Paasche = current quantities (Qₙ), Fisher = geometric mean of both. Tip 3: For the Time Reversal Test, compute the index both ways and verify their product equals 1. Tip 4: If asked which index is “best,” Fisher’s Ideal is the answer — explain it satisfies both TRT and FRT. Tip 5: For CPI questions, know the formula: Inflation = [(New CPI − Old CPI)/Old CPI] × 100. Tip 6: Chain base index = (current year’s link relative × previous year’s chain index) / 100.