Distinguish between fixed and chain based index numbers

Fixed-base index numbers and chain-based index numbers are two common methods used in economics and statistics to measure changes in the value of a variable over time. Here's a brief distinction between the two:

1. Fixed-base Index Numbers:
   - In fixed-base index numbers, a specific period is chosen as the base period against which subsequent periods are compared.
   - The prices or quantities of items in the base period are given a fixed weight of 100, and changes in subsequent periods are expressed relative to this base period.
   - The formula for calculating a fixed-base index is typically: 
     \[ \text{Index} = \left( \frac{\text{Value in Current Period}}{\text{Value in Base Period}} \right) \times 100 \]
   - Fixed-base indices are easy to calculate and understand, but they can become outdated over time if the base period no longer accurately represents the current economy.

2. **Chain-based Index Numbers:**
   - Chain-based index numbers do not rely on a fixed base period; instead, they use a series of linked index numbers to measure changes over time.
   - In a chain index, each period is compared to the preceding period, rather than to a fixed base period.
   - Chain indices are often preferred when the characteristics of the economy change significantly over time, as they can adapt to these changes without needing to update a fixed base period.
   - The Laspeyres and Paasche methods are commonly used to calculate chain-based indices. The Laspeyres method uses quantities from the base period, while the Paasche method uses quantities from the current period.
   - Chain-based indices provide a more accurate representation of changing economic conditions but may be more complex to calculate and interpret.

In summary, fixed-base index numbers use a single base period for comparison, while chain-based index numbers use linked periods to measure changes over time. Each method has its advantages and disadvantages, and the choice between them depends on factors such as the stability of the economy and the desired level of accuracy in the index measurement.