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