Index Number Properties — A First Look
Imagine you want to know whether your pocket money has kept up with rising prices. You remember that last year a plate of biryani cost ₹100; today it costs ₹120. That single item tells you prices went up 20%. But what if you also buy notebooks, bus tickets, and chai? Some things rose faster, some slower, maybe one even fell. How do you combine all those changes into one number that honestly reflects "the price level"? That number is an index number — and the properties we are about to discuss are the rules that keep it honest.
What is an index number, really?
An index number is a single figure that shows the relative change in a variable (or a group of variables) from one time period to another. The most famous one you will meet is the Consumer Price Index (CPI), which tracks the cost of a fixed basket of goods and services. If the CPI rises from 100 to 110, it means the same basket now costs 10% more — your money buys less.
Price Index=∑p0q0∑p1q0×100
where p0 = price in base year, p1 = price in current year, q0 = quantity in base year (fixed basket). This is the Laspeyres formula, the one most commonly used in India's CPI.
But a number is only useful if it behaves sensibly. That is where properties come in.
The three essential properties
1. Unit Test (or Commodity Reversal Test)
The index number should not change if you simply change the unit in which you measure a commodity. Suppose you measure rice in kilograms — the index should be the same as if you measured it in grams (just scaled). This sounds obvious, but not every formula passes it. The Laspeyres and Paasche indices pass this test because they use ratios of sums, and changing units multiplies numerator and denominator by the same factor, which cancels out.
A common mistake: thinking the unit test means the index stays the same when you swap which year is base. That is a different property (time reversal). Keep them separate.
2. Time Reversal Test
If you compute the index from year A to year B, and then from year B back to year A, the product of the two should equal 1 (or 100, if using percentage form). In symbols:
I01×I10=1
Why does this matter? Because if you go forward and then backward, you should end up exactly where you started — no "phantom" gain or loss. The Laspeyres index fails this test. The Paasche index also fails it. Only the Fisher Ideal Index (the geometric mean of Laspeyres and Paasche) passes the time reversal test.
For exams: remember that Fisher's index is called "ideal" precisely because it satisfies both the time reversal and factor reversal tests. Laspeyres and Paasche each fail one.
3. Factor Reversal Test
This is the trickiest one. The idea: if you multiply a price index by a quantity index, you should get the value index (the ratio of total expenditure). In symbols:
Price Index×Quantity Index=∑p0q0∑p1q1
Think of it this way: total money spent = price × quantity. So if your price index says "prices rose 10%" and your quantity index says "quantity bought rose 5%", then total spending should have risen by about 15.5% (1.10 × 1.05 = 1.155). The factor reversal test demands this relationship hold exactly. Again, only the Fisher index passes this test.
Why should you care about these properties?
In the real world, index numbers are used to adjust salaries, pensions, and government benefits. If the index is biased — say it overstates inflation — then pensioners get more money than they need, and the government wastes funds. If it understates inflation, workers lose purchasing power. The properties are not abstract math; they are safeguards against misleading numbers.
For your Class 12 board exam, you will most often be asked to: …