Index Number Properties — The Intuition First
Imagine you're comparing the price of a basket of goods across two years. You collect data, compute an index number — say 120 for 2024 with 2020 as base. That number tells you prices have risen 20%. But here's the catch: the number you get depends on how you computed it. Did you use the old quantities or the new ones? Did you average the prices first or the ratios? Different methods give different answers.
That's where index number properties come in. They are a set of logical tests — like a checklist — that a good index number should satisfy. If an index fails too many of these tests, it's probably misleading.
The Core Idea
An index number is a function that takes two sets of data (prices and quantities for a base period and a current period) and returns a single number. The properties are criteria that this function should meet to be considered "well-behaved." Think of them as the rules of fair play for index numbers.
No single index number satisfies all properties perfectly. The properties help you choose the right index for your purpose and understand its limitations.
The Key Properties — One by One
1. Identity Test
If the current period is the same as the base period, the index should be 100 (or 1, depending on scale). Obvious, but essential.
Ia,a=100
2. Proportionality (or Homogeneity) Test
If all prices in the current period are multiplied by a constant k, the index should also be multiplied by k. For example, if every price doubles, the index should double (from 100 to 200).
I(kp1,p0,q)=k⋅I(p1,p0,q)
3. Time Reversal Test
This is a subtle but powerful idea. If you swap the base and current periods, the product of the two indices should equal 1 (or 100 × 100 = 10000 if using percentages). In other words, the index going forward and the index going backward should be reciprocals.
I0,1×I1,0=1
Laspeyres and Paasche indices fail this test. Fisher's Ideal Index passes it — that's why Fisher's is often preferred.
4. Factor Reversal Test
This is the most demanding test. It says that if you compute a price index and a quantity index using the same formula, their product should equal the ratio of total expenditure (value) in the two periods.
P01×Q01=∑p0q0∑p1q1
Only Fisher's Ideal Index passes this test.
5. Circular Test
This extends the time reversal idea to three periods. If you compute indices from period 0 to 1, 1 to 2, and 0 to 2, the product of the first two should equal the third.
I0,1×I1,2=I0,2
Almost all common indices fail this test. It's considered too restrictive for practical use.
A Quick Reference Table
| Property | What it checks | Laspeyres | Paasche | Fisher |
|---|
| Identity | Ia,a=100 | Pass | Pass | Pass |
| Proportionality | I(kp)=kI(p) | Pass | Pass | Pass |
| Time Reversal | I0,1×I1,0=1 | Fail | Fail | Pass |
| Factor Reversal | P×Q=value ratio | Fail | Fail | Pass |
| Circular | I0,1×I1,2=I0,2 | Fail | Fail | Fail |
Why Should You Care?
When you're solving exam problems, you'll often be asked to "check whether this index satisfies the time reversal test." That's a mechanical calculation. But the real point is deeper: these properties reveal the bias in an index.
Laspeyres tends to overstate price rises (because it uses old quantities, ignoring substitution). Paasche tends to understate them. Fisher, by taking the geometric mean, balances the two — and its ability to pass both reversal tests is why it's called "ideal."
For exams: memorize which indices pass which tests. Fisher passes both reversal tests. Laspeyres and Paasche fail both. No common index passes the circular test.
The Bottom Line
Index number properties are not abstract math — they are a practical toolkit. They tell you: If I use this index, what am I assuming? What am I ignoring? A good index is not just one that gives a number; it's one whose number you can trust. The properties are your trust checklist.