Introduction: Since the partial derivative of a function of two variables is defined as the derivative of a function of one variable, we can easily define the notion of elasticity for a function of two variables with respect to each of the variables separately.
Definition: The partial elasticity of the function $z(x,y)$ with respect to the variable $x$ is denoted by
\[
\epsilon_x=z_x'(x,y)\cdot \frac{x}{z(x,y)}.
\]
The partial elasticity of the function $z(x,y)$ with respect to the variable $y$ is denoted by
\[
\epsilon_y=z_y'(x,y)\cdot \frac{y}{z(x,y)}.
\]
Property: For a percentage change $\%\Delta z$ of the function value $z(x,y)$, caused by a small percentage change $\% \Delta x$ of the variable $x$ at a constant value of the variable $y$, we have
\[
\%\Delta z\approx \epsilon_x \cdot \%\Delta x.
\]
For a percentage change $\%\Delta z$ of the function value $z(x,y)$, caused by a small percentage change $\% \Delta y$ of the variable $y$ at a constant value of the variable $x$, we have
\[
\%\Delta z\approx \epsilon_y \cdot \%\Delta y.
\]
Definition: The partial elasticity of the function $z(x,y)$ with respect to the variable $x$ is denoted by
\[
\epsilon_x=z_x'(x,y)\cdot \frac{x}{z(x,y)}.
\]
The partial elasticity of the function $z(x,y)$ with respect to the variable $y$ is denoted by
\[
\epsilon_y=z_y'(x,y)\cdot \frac{y}{z(x,y)}.
\]
Property: For a percentage change $\%\Delta z$ of the function value $z(x,y)$, caused by a small percentage change $\% \Delta x$ of the variable $x$ at a constant value of the variable $y$, we have
\[
\%\Delta z\approx \epsilon_x \cdot \%\Delta x.
\]
For a percentage change $\%\Delta z$ of the function value $z(x,y)$, caused by a small percentage change $\% \Delta y$ of the variable $y$ at a constant value of the variable $x$, we have
\[
\%\Delta z\approx \epsilon_y \cdot \%\Delta y.
\]