Advanced Microeconomics/Homogeneous and Homothetic Functions

Homogeneous & Homothetic Functions

For any scalar $k$ a function is homogenous if $f(t{x}_1,t{x}_2,\dots ,t{x}_n)=t^{k}f(x_1,x_2,\dots ,x_n)$ 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 $\mathrm{\forall }x,y\in {ℝ}^{n}x\ge y\to f(x)\ge f(y)$
• 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

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