Advanced Microeconomics/Homogeneous and Homothetic Functions

For any scalar ${\displaystyle k}$ a function is homogenous if ${\displaystyle f(tx_{1},tx_{2},\dots ,tx_{n})=t^{k}f(x_{1},x_{2},\dots ,x_{n})}$ A homothetic function is a monotonic transformation of a homogeneous function, if there is a monotonic transformation ${\displaystyle g(z)}$ and a homogenous function ${\displaystyle h(x)}$ such that f can be expressed as ${\displaystyle g(h)}$

• A function is monotone where ${\displaystyle \forall \;x,y\in \mathbb {R} ^{n}\;x\geq y\rightarrow f(x)\geq f(y)}$
• Assumption of homotheticity simplifies computation,
• Derived functions have homogeneous properties, doubling prices and income doesn't change demand, demand functions are homogenous of degree 0
• The slope of the MRS is the same along rays through the origin

${\displaystyle {\begin{aligned}Q&=x^{\frac {1}{2}}y^{\frac {1}{2}}+x^{2}y^{2}\\&{\mbox{Q is not homogeneous, but represent Q as}}\\&g(f(x,y)),\;f(x,y)=xy\\g(z)&=z^{\frac {1}{2}}+z^{2}\\g(z)&=(xy)^{\frac {1}{2}}+(xy)^{2}\\&{\mbox{Calculate MRS,}}\\{\frac {\frac {\partial Q}{\partial x}}{\frac {\partial Q}{\partial y}}}&={\frac {{\frac {\partial Q}{\partial z}}{\frac {\partial f}{\partial x}}}{{\frac {\partial Q}{\partial z}}{\frac {\partial f}{\partial y}}}}={\frac {\frac {\partial f}{\partial x}}{\frac {\partial f}{\partial y}}}\\&{\mbox{the MRS is a function of the underlying homogenous function}}\end{aligned}}}$