An index number is a statistical measure designed to show the relative change in a variable or a group of variables with respect to time, geographic location or other characteristic.
Spiegel: "An index number is a number that measures the relative change in a set of measurements over time."
The reference period is called the base period (index taken as 100); the period of comparison is called the current period.
Using geometric mean of relatives instead: \[ P_{01} = \text{antilog}\!\left[\dfrac{1}{n}\sum \log(p_1/p_0)\right] \times 100. \]
Prices of 4 items: \(p_0\) = 5, 8, 10, 12; \(p_1\) = 6, 10, 14, 15.
Simple aggregate: \(\sum p_1/\sum p_0 \times 100 = 45/35 \times 100 = 128.57\).
Simple AM of relatives: \(1/4 (120 + 125 + 140 + 125) = 127.5\).
Wages of 3 categories rose: \(p_0\) = 200, 300, 500; \(p_1\) = 240, 360, 600. Aggregate index = (240+360+600)/(200+300+500) × 100 = 1200/1000 × 100 = 120. (20 % rise.)
Uses fixed base-year basket → easier to compute, but tends to over-estimate inflation (consumers substitute away from expensive items).
Uses current basket → reflects current consumption, but tends to under-estimate inflation.
Average of base and current quantities used as weights.
Geometric mean of Laspeyres and Paasche. Called "ideal" because it satisfies both the time-reversal and factor-reversal tests.
Arithmetic mean of Laspeyres and Paasche.
where \(q\) is a fixed weight chosen by the investigator (often \(q = (q_0 + q_1)/2\)).
Data:
| Item | p₀ | q₀ | p₁ | q₁ |
|---|---|---|---|---|
| A | 4 | 10 | 6 | 12 |
| B | 5 | 15 | 8 | 14 |
| C | 10 | 5 | 12 | 4 |
Compute \(\sum p_0 q_0 = 40+75+50 = 165;\; \sum p_1 q_0 = 60+120+60 = 240;\; \sum p_0 q_1 = 48+70+40 = 158;\; \sum p_1 q_1 = 72+112+48 = 232\).
Laspeyres: 240/165 × 100 = 145.45.
Paasche: 232/158 × 100 = 146.84.
Marshall–Edgeworth: (240 + 232)/(165 + 158) × 100 = 472/323 × 100 = 146.13.
Fisher: \(\sqrt{145.45 \times 146.84} = \sqrt{21358.0} = \) 146.14.
Bowley: (145.45 + 146.84)/2 = 146.15.
If \(P^L = 110\) and \(P^P = 108\), Fisher = \(\sqrt{110 \times 108} = \sqrt{11\,880} = 108.99\); Bowley = 109.
A good index should reverse itself when base and current periods are interchanged.
The product of price and quantity indices should equal the value index.
Indices should be consistent for chained transitions through multiple periods.
| Index | TRT | FRT | Circular |
|---|---|---|---|
| Laspeyres | No | No | No |
| Paasche | No | No | No |
| Marshall–Edgeworth | Yes | No | No |
| Fisher (Ideal) | Yes | Yes | No |
| Kelly (fixed weights) | Yes | No | Yes |
Fisher's index satisfies TRT and FRT (hence "ideal") but not the circular test.
The Cost of Living Index (or Consumer Price Index, CPI) measures changes in the cost of a fixed basket of goods and services consumed by a target population (e.g., industrial workers, agricultural labour, urban consumers).
where \(W = p_0 q_0\) is the weight (base-year expenditure) for each item.
4 items with weights 30, 25, 20, 25 (sum 100); price relatives 110, 120, 125, 130.
CPI = (110 × 30 + 120 × 25 + 125 × 20 + 130 × 25)/100 = (3300 + 3000 + 2500 + 3250)/100 = 12 050/100 = 120.5.
Money wage = ₹15 000; CPI = 150. Real wage = 15 000/150 × 100 = ₹10 000 (purchasing power at base prices).
The Wholesale Price Index measures changes in the average wholesale prices of a fixed basket of commodities (primary articles, fuel & power, manufactured products). Used to estimate inflation at the producer level.
| Aspect | CPI | WPI |
|---|---|---|
| Level | Retail (consumer) | Wholesale (producer) |
| Basket | Goods + services | Goods only |
| Includes services? | Yes | No |
| Target user | Households | Industry, govt. |
| Used for | DA, real wages, RBI's inflation target | Producer-price trends, deflating output |