# Measuring quantities for price indices?

I'm confused about how one acquires the data necessary to compute price indices--specifically, how one acquires quantity data.

Suppose I wanted to compute the Laspeyres Index, defined as

$\sum\limits_{i=1}^{N} p_{i}(t_{f}) q_{i}(t_{0}) / \sum\limits_{i=1}^{N} p_{i}(t_{0}) q_{i}(t_{0})$

where $N$ is the number of different goods, each $p_{i}$ is a price, and each $q_{i}$ is a quantity. Each $p_i$ is measured in units of currency per units of a good, and each $q_{i}$ is measured in units of a good per unit of time.

But as far as I can tell, we can't measure $q_{i}(t_{o})$ or $q_{i}(t_{f})$. These represent the instantaneous rates that a certain good is sold at a particular time, whereas data-collectors are only able to ask how much firms have sold of a good over a period of time--in other words, we can measure

$\int\limits_{t_{0}}^{t_{f}} q_{i}(t)dt$

by summing the total sales of all firms that sell good $i$ during the period between $t_{0}$ and $t_{f}$, but not

$q_{i}(t)$

for any given $t$.

How do actual econometricians measure $q_{i}(t_{o})$ and $q_{i}(t_{f})$? Do they measure the number of goods sold $x$ days after $t_{0}$ or $t_{f}$, then divide by $x$ for a sort of local approximation? Or do they use some other method?

• After answering your question I became uncertain. Is $q_i(t)$ how you think econometricians actually measure quantitites or is it how you want to measure quantities? – Giskard Dec 30 '17 at 10:15
• @denesp Writing quantity as a function of time and indexing quantity to time are the same thing mathematically--I wrote them this way because I thought it would be easier to read, I'm sorry if this caused more confusion than it cleared up. I've also seen function notation used in the context of Divisia indices, which (most) other price indices are approximations to. I assumed that since the data for Divisia indices are measured in continuous time, other price indices simply used individual values of $q_{i}(t)$, rather than the complete set of values needed to compute the Divisia index. – SilasLock Jan 1 '18 at 23:25
• From the Wikipedia page for Divisia indices: "In practice, economic data are not measured in continuous time." – Giskard Jan 1 '18 at 23:32

The notation $q_{i}(t_{0})$ does not necessarily mean number of goods sold in instant $t_0$. It can also mean the number of goods sold in the period denoted by $t_0$. As you point out the first one would be somewhat difficult to measure.
In fact I have not yet seen the notation you use. The notation I am familiar with is usually something like $q_{i,t_0}$, which better indicates that time is treated as a discrete variable. The unit of time could be anything, but it is usually a year or quarter.