Laspeyres and Paasche Indices

Tom Crossley taught a course on index number theory and got me fascinated with the subject.

Laspeyres is probably the most well known price index and I will mostly focus on it instead of Paasche in this post. Suppose you have baseline (denoted by “0” subscript) and current (denoted by “1” subscript) prices, quantities and utilities. I’ll discuss briefly below how Laspeyres provides a bound to baseline utility COLI (Cost of Living Index).

Define the Laspeyres Price Index as:

P_{LA}=\frac{\sum^{p_1*q_0}}{\sum^{p_0*q_0}} = \frac{\sum^{p_1*q_0}}{X_0}}

Then COLI at baseline utility is:

\frac{c(p_1,u_0)}{c(p_0, u_0)}=\frac{c(p_1,u_0)}{X_0}

Since c(p_1,u_0) is the least expensive way to obtain baseline utility at current prices and \sum^{p_1*q_0} is a way of obtaining baseline utility at current prices it follows that the former is smaller or equal than the latter. Therefore, Laspeyres is an upper bound to COLI, evaluated at baseline utility. Analogously follows that Paasche provides a lower bound for (current) COLI.

As Tom recently pointed out to me, it’s important to note that COLI at baseline is not equal to current, unless, you assume homothetic preferences. The downside is that homothetic preferences imply that regardless of income, households have the same spending patterns. Yet, if you’re willing to assume that, then Laspeyres and Paasche end up providing bounds to the same COLI (since baseline and current COLI would be equal). Otherwise, not.