# Example of formula bias in CPI

A macroeconomics textbook explains the formula bias in the consumer price index (CPI) in the following way:

This bias results because price data are collected on a disaggregated basis and then aggregated in a very complex manner that can introduce anomalies. For example, the calculation method used in recent years gives too much weight to items on sale; somewhat paradoxically, this generates formula-induced inflation as the items go off sale. The degree of this bias can increase with the frequency of rotation (of outlets included in the sample), because the bias results from short-run price variability and the use of a method that gives greater weight to lower-than-average prices.

I guess it describes a situation like this (it's not from the book. I made it up):

Walmart sells apples on Monday. On Tuesday, it sells bananas but no apples anymore. On Wednesday, it sells oranges but no apples or bananas. This pattern repeats weekly.

Firstly, I'm not sure how giving too much weight to Monday's apples generates inflation. I thought the CPI value would be decreased when the apples "go off sale" because the quantity sold drops.

Secondly, I'm not sure how giving greater weight to "lower-than-average prices," in particular, would increase the degree of bias.

Since no other fruit was sold on Monday, the CPI will only be constructed from apples, and it will also be our base period, so CPI is just 100. However, on Tuesday, the CPI will be all composed of bananas, which will be 200, and the CPI on Wednesday will be 500. Then when we calculate inflation $$(CPI_{t+1}-CPI_t)/CPI_t$$, Tuesday inflation will be 100% and Wednesday inflation 150%.