Advanced Microeconomics/Homogeneous and Homothetic Functions

Homogeneous & Homothetic Functions
For any scalar $$k$$ a function is homogenous if $$f(tx_1,tx_2,\dots,tx_n)=t^kf(x_1,x_2,\dots,x_n) $$ 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
$$ \begin{align} Q &= x^{\frac{1}{2}}y^{\frac{1}{2}}+ x^2y^2\\ &\mbox{Q is not homogeneous, but represent Q as}\\ &g(f(x,y)), \; f(x,y) = xy \\ g(z) &= z^{\frac{1}{2}}+ z^2\\ g(z) &= (xy)^{\frac{1}{2}} + (xy)^2\\ &\mbox{Calculate MRS,}\\ \frac{\frac{\partial Q}{\partial x}}{\frac{\partial Q}{\partial y}} &= \frac{\frac{\partial Q}{\partial z} \frac{\partial f}{\partial x}}{\frac{\partial Q}{\partial z} \frac{\partial f}{\partial y}} = \frac{ \frac{\partial f}{\partial x}}{  \frac{\partial f}{\partial y}}\\ &\mbox{the MRS is a function of the underlying homogenous function} \end{align} $$