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:

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

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.