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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes
TECHNICAL NOTE: A MATHEMATICAL APPROACH TO PRICE INDEXES
Notation, Laspeyres, and Paasche Indexes
We start by introducing some notation for the variables that most concern us, prices and quantities. In each period t, there are N goods, each of which has a price, pn, and a quantity, qn, with the subscript n labeling the good and running from 1 to N. We shall also need to refer to these prices and quantities in different periods, typically a base or reference period, denoted 0, and a later comparison or current period, denoted t. (We will occasionally separate base and reference periods later.) Superscripts refer to these time periods, so that is the purchase of good n in period t. Sometimes we need to distinguish between purchases by different people, in which case we add another superscript h, for household. It is also occasionally useful to use vector notation, in which case subscripts are dropped; hence q is the (column) vector of N quantities and p the corresponding vector of N prices. Associated with the vector notation is the “dot” or inner product, p . q, which denotes the sum of the element by element product of the vectors, in this case the total amount of money spent on q when it is bought at prices p.
Armed with only this notation, we can introduce the two most important fixed-basket price indexes or cost-of-goods indexes, or COGIs. For the Laspeyres price index, there is a base set of quantities, which we can denote q0, which is repriced in successive periods. Hence, the Laspeyres price index for period t, which we denote is defined by the equation
In equation (1), the two sets of prices pt and p0 are compared using the base period quantities, q0, as weights. Note that the numerator and denominator of (1) are identical, except that the prices in the numerator are current prices pt, while those in the denominator are base period prices p0. A useful alternative way to write the Laspeyres index is to define a price relative for each good. We write for good n
which can be used to rewrite equation (1) in the form