Advanced Microeconomics/Homogeneous and Homothetic Functions

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Homogeneous & Homothetic Functions[edit]

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:[edit]


\begin{align}
Q &= x^{\frac{1}{2}}y^{\frac{1}{2}}+ x^2y^2\\
&\mbox{Q is not homogenous, 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}