Advanced Microeconomics/Homogeneous and Homothetic Functions

Homogeneous & Homothetic Functions

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

A function is monotone where $\forall \;x,y\in \mathbb {R} ^{n}\;x\geq y\rightarrow f(x)\geq f(y)$
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

Example

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

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