I hadn't realised until now that the Johnson review has a nice box on Chain drift.
Here's the idea.
1. Suppose the item 1 is detergent
2. "The price of item 1 halves between periods 1 and 2, causing a big jump in sales. The product returns to its original price in period 3 and sales go back to original share"
3. Good 2 remains same price
4. "Most people would agree that the price index in period 1 should be the same as period 4; after all, the price and quantity sold of both products is the same."
5. And the Laspeyes index delivers that.
6. here's teh problem. the chained index is "calculates overall price change between each pair of periods in sequence and then multiplies them together. So the chained price index in period 4 is obtained by multiplying the price changes in periods 1-2, 2-3 and 3-4 together.
7. As you can see, the chained index does NOT return to its original value.
a. in the chained Laspeyres index, chain drift arises because the product 1 price fall between periods 1 and 2 has low weight (only 10 of product 1 were bought), while the increase to period 3 has high weight (5,000 products).
b. In the chained Törnqvist index, the average expenditure share (which is what matters in the Törnqvist) is larger in periods 1 and 2 when the price is falling, than when the price recovers between periods 2 and 3.
Here's some working: this is a problem for scanner data.
