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
ϵx=z′x(x,y)⋅xz(x,y).
The partial elasticity of the function z(x,y) with respect to the variable y is denoted by
ϵy=z′y(x,y)⋅yz(x,y).
Property: For a percentage change %Δz of the function value z(x,y), caused by a small percentage change %Δx of the variable x at a constant value of the variable y, we have
%Δz≈ϵx⋅%Δx.
For a percentage change %Δz of the function value z(x,y), caused by a small percentage change %Δy of the variable y at a constant value of the variable x, we have
%Δz≈ϵy⋅%Δy.
Definition: The partial elasticity of the function z(x,y) with respect to the variable x is denoted by
ϵx=z′x(x,y)⋅xz(x,y).
The partial elasticity of the function z(x,y) with respect to the variable y is denoted by
ϵy=z′y(x,y)⋅yz(x,y).
Property: For a percentage change %Δz of the function value z(x,y), caused by a small percentage change %Δx of the variable x at a constant value of the variable y, we have
%Δz≈ϵx⋅%Δx.
For a percentage change %Δz of the function value z(x,y), caused by a small percentage change %Δy of the variable y at a constant value of the variable x, we have
%Δz≈ϵy⋅%Δy.