对于任何标量 k {\displaystyle k} ，如果函数满足 f ( t x 1 , t x 2 , … , t x n ) = t k f ( x 1 , x 2 , … , x n ) {\displaystyle f(tx_{1},tx_{2},\dots ,tx_{n})=t^{k}f(x_{1},x_{2},\dots ,x_{n})} ，则该函数是齐次函数。如果存在一个单调变换 g ( z ) {\displaystyle g(z)} 和一个齐次函数 h ( x ) {\displaystyle h(x)} ，使得 f 可以表示为 g ( h ) {\displaystyle g(h)} ，则该函数是同质函数。
Q = x 1 2 y 1 2 + x 2 y 2 Q is not homogeneous, but represent Q as g ( f ( x , y ) ) , f ( x , y ) = x y g ( z ) = z 1 2 + z 2 g ( z ) = ( x y ) 1 2 + ( x y ) 2 Calculate MRS, ∂ Q ∂ x ∂ Q ∂ y = ∂ Q ∂ z ∂ f ∂ x ∂ Q ∂ z ∂ f ∂ y = ∂ f ∂ x ∂ f ∂ y the MRS is a function of the underlying homogenous function {\displaystyle {\begin{aligned}Q&=x^{\frac {1}{2}}y^{\frac {1}{2}}+x^{2}y^{2}\\&{\mbox{Q is not homogeneous, but represent Q as}}\\&g(f(x,y)),\;f(x,y)=xy\\g(z)&=z^{\frac {1}{2}}+z^{2}\\g(z)&=(xy)^{\frac {1}{2}}+(xy)^{2}\\&{\mbox{Calculate MRS,}}\\{\frac {\frac {\partial Q}{\partial x}}{\frac {\partial Q}{\partial y}}}&={\frac {{\frac {\partial Q}{\partial z}}{\frac {\partial f}{\partial x}}}{{\frac {\partial Q}{\partial z}}{\frac {\partial f}{\partial y}}}}={\frac {\frac {\partial f}{\partial x}}{\frac {\partial f}{\partial y}}}\\&{\mbox{the MRS is a function of the underlying homogenous function}}\end{aligned}}}
，如果函数满足 
，则该函数是齐次函数。如果存在一个单调变换 
和一个齐次函数 
，使得 f 可以表示为 
，则该函数是同质函数。
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