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 |
|----------|----------------|-----------|---------|--------| …