Cross-price elasticity of demand

Definition

Suppose $A$ and $B$ are two commodities. The cross-price elasticity of demand of $A$ with respect to $B$ measures the fractional change in the demand of $A$ in response to a fractional change in the unit price of $B$ . Note that the price of $A$ is not changed in the process.

Formally, if $p_{A}$ and $p_{B}$ denote the unit prices of $p_{A}$ and $p_{B}$ and $q_{A}$ and $q_{B}$ denote the quantities demanded for $A$ and $B$ , the cross-price elasticity is given by any of the following four equivalent formulations:

$\frac{p_{B} d q_{A}}{q_{A} d p_{B}} = \frac{d q_{A} / q_{A}}{d p_{B} / p_{B}} = \frac{d q_{A} / d p_{B}}{q_{A} / p_{B}} = \frac{d (\log (q_{A}))}{d (\log (p_{B}))}$

Note that although only $q_{A}$ and $p_{B}$ appear in the expression for cross-price elasticity, the value $p_{A}$ could also affect the value of cross-price elasticity. More specifically, the formula makes sense against a backdrop of the value of $p_{A}$ and all the other determinants of demand.