The axioms

There are five key axioms that any price index should satisfy for all collections of prices and quantities. These axioms can be seen as generalizing the properties of the ratio of prices between period \(t\) and period 0 for a single good or service to settings with many goods and services.

  1. Monotonicity \(I(p'_{t}, p_{0}, q_{t}, q_{0}) > I(p_{t}, p_{0}, q_{t}, q_{0})\) if \(p'_{t} \geq p_{t}\) and \(p'_{t} \neq p_{t}\), and \(I(p_{t}, p'_{0}, q_{t}, q_{0}) < I(p_{t}, p_{0}, q_{t}, q_{0})\) if \(p'_{0} \geq p_{0}\) and \(p'_{0} \neq p_{0}\).

  2. Linear homogeneity \(I(cp_{t}, p_{0}, q_{t}, q_{0}) = cI(p_{t}, p_{0}, q_{t}, q_{0})\) for any \(c > 0\).

  3. Identity \(I(p_{0}, p_{0}, q_{t}, q_{0}) = 1\).

  4. Homogeneity of degree zero \(I(cp_{t}, cp_{0}, q_{t}, q_{0}) = I(p_{t}, p_{0}, q_{t}, q_{0})\) for any \(c > 0\).

  5. Dimensional invariance If \(A\) is a diagonal matrix of positive numbers, then \(I(Ap_{t}, Ap_{0}, A^{-1}q_{t}, A^{-1}q_{0}) = I(p_{t}, p_{0}, q_{t}, q_{0})\).

All of these axioms are fairly straightforward to understand from their mathematical representation, perhaps with the exception of dimensional invariance, and are quite intuitive. Monotonicity simply means that a price index is increasing in period \(t\) prices and decreasing in period 0 prices—larger period \(t\) prices produce larger index values and larger period 0 prices produce smaller index values. This seems like a necessary condition for a price index to meaningfully measure inflation, as an index that does not satisfy monotonicity can decrease when prices increase.

Linear homogeneity is probably the least intuitive of the axioms, and says that multiplying all period \(t\) prices by a constant is the same as multiplying the entire index by a constant, so that a proportional increase in prices results in a proportional increase in the index. To see why this makes sense, consider two cities (A and B) that have all the same prices in period 0. If all prices in city A increase by twice as much as prices in city B between period 0 and period \(t\), with quantities purchased remaining the same in the two cities (because relative prices are the same in both cities), then it is reasonable to say that prices in city A have increased by twice as much as in city B.

The identity axiom is simple, and states that a price index does not show a change in prices if prices do not change. Like the monotonicity axiom, this seems like a necessary requirement for a price index to measure inflation; otherwise, a price index could show a change in prices when prices do not change over time.

Homogeneity of degree zero means that multiplying all prices by a constant has no impact on an index—the currency that prices are measured in does not affect the value of a price index. An index-number formula that does not satisfy homogeneity of degree zero can give a different measure of inflation depending on the currency prices are measured in, and cannot be used to compare inflation in different countries.

Dimensional invariance looks complex, but it simply says that changing the units of measurement does not change the index value. A price index should not change if all prices are multiplied by a constant and all quantities are divided by the same constant—a price index should not depend on the units of measurement.

As an example, it is straightforward to see that the Laspeyres index satisfies all five axioms.

  1. For monotonicity, \[\begin{align*} I(p'_{t}, p_{0}, q_{t}, q_{0}) = \frac{\sum_{i=1}^{n} p'_{it}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} > \frac{\sum_{i=1}^{n} p_{it}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} = I(p_{t}, p_{0}, q_{t}, q_{0}) \end{align*}\] whenever each \(p'_{it} \geq p_{it}\), with at least one strictly greater. The opposite holds if each \(p'_{i0} \geq p_{i0}\), with at least one strictly greater.

  2. For linear homogeneity, \[\begin{align*} I(cp_{t}, p_{0}, q_{t}, q_{0}) = \frac{\sum_{i=1}^{n} cp_{it}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} = \frac{c\sum_{i=1}^{n} p_{it}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} = cI(p_{t}, p_{0}, q_{t}, q_{0}) \end{align*}\] for any \(c > 0\).

  3. For identity, \[\begin{align*} I(p_{0}, p_{0}, q_{t}, q_{0}) = \frac{\sum_{i=1}^{n} p_{i0}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} = 1. \end{align*}\]

  4. For homogeneity of degree zero, \[\begin{align*} I(cp_{t}, cp_{0}, q_{t}, q_{0}) = \frac{\sum_{i=1}^{n} cp_{it}q_{i0}}{\sum_{i=1}^{n} cp_{i0}q_{i0}} = \frac{c\sum_{i=1}^{n} p_{it}q_{i0}}{c\sum_{i=1}^{n} p_{i0}q_{i0}} = I(p_{t}, p_{0}, q_{t}, q_{0}) \end{align*}\] for any \(c > 0\).

  5. For dimensional invariance, if \[\begin{align*} A = \begin{bmatrix} a_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & a_n \end{bmatrix} \end{align*}\] then \[\begin{align*} I(Ap_{t}, Ap_{0}, A^{-1}q_{t}, A^{-1}q_{0}) = \frac{\sum_{i=1}^{n} a_{i}p_{it}a_{i}^{-1}q_{i0}}{\sum_{i=1}^{n} a_{i}p_{i0}a_{i}^{-1}q_{i0}} = \frac{\sum_{i=1}^{n} p_{it}q_{i0}}{\sum_{i=1}^{n} p_{i0}q_{i0}} = I(p_{t}, p_{0}, q_{t}, q_{0}). \end{align*}\]

It should be fairly safe to say that any sensible price index should satisfy these axioms. Although there are many price indices that do, the axioms work to exclude unreasonable index-number formulas that make no sense for measuring changes in prices.