Laspeyeres and Paasche as bounds of COLI

Tom Crossley taught a course on index number theory and got me fascinated with the subject. This post essentially summarizes how 2 price indices provide bounds to a cost of living index (COLI).

Suppose you have baseline (denoted by a “0” subscript) and current (denoted by a “1” subscript) prices, quantities and utilities.

Define the Laspeyeres 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 base-line 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, Laspeyeres is an upper bound to COLI, evaluated at baseline utility.

Analogously follows that Paasche provides a lower bound for COLI.