10.1 Index Numbers
An index number measures the relative change in a quantity (such as price, quantity, or value) at a specific time compared to a base year.
Price Index Formula
The price index $I$ at time $t_1$ relative to base time $t_0$ is given by:
$$I = \frac{Q_1}{Q_0} \times 100$$
where:
- $Q_1$ = Price or quantity at the specific time (current year/time $t_1$).
- $Q_0$ = Price or quantity at the base time (base year/time $t_0$).
Interpretation of Index Numbers
- $I = 100$: No change in price compared to the base year.
- $I > 100$: Price has increased by $(I - 100)\%$ compared to the base year.
- $I < 100$: Price has decreased by $(100 - I)\%$ compared to the base year.
10.2 Composite Index
A composite index ($\bar{I}$) combines several individual index numbers, where each index number is weighted according to its relative importance.
Composite Index Formula
$$\bar{I} = \frac{\sum I_i w_i}{\sum w_i}$$
where:
- $I_i$ = Price index of component $i$.
- $w_i$ = Weightage of component $i$ (can be represented as percentages, ratios, frequencies, degrees in a pie chart, etc.).
Changes in Base Year or Multi-Year Index Transformations
When chaining index numbers across multiple years (e.g., $t_0$, $t_1$, and $t_2$):
$$I_{t_2/t_0} = \frac{I_{t_1/t_0} \times I_{t_2/t_1}}{100}$$
10.3 Application of Index Numbers
Index numbers are widely used in economics and real-world budgeting to analyze trends such as:
- Consumer Price Index (CPI): Measures the average changes in prices paid by consumers for goods and services.
- Cost of Living Index: Evaluates changes in household expenses over time.
- Finding overall cost or expenditure changes based on future projections.