Homogeneous and Homothetic Function

Homogeneous function

A homogeneous function is a special type of function in which the scaling of inputs leads to a proportional scaling of outputs. In simpler terms, if inputs increase by $\lambda$ fold, then output increases by $\lambda$ fold or some fold of $\lambda$.

Mathematically,

$f(x,y)=xy---(i)$

Suppose that equation (i) is a utility function and x and y represent good X and good Y. Now, if good X and good Y is increased by $\lambda$ fold, then

$f({\lambda}x,{\lambda}y)=({\lambda}x)({\lambda}y)$

$f({\lambda}x,{\lambda}y)={\lambda}^2xy$  

$f({\lambda}x,{\lambda}y)={\lambda}^2f(x,y)---(ii)$

From equation (ii), we conclude that

$f(x,y)$ is a homogeneous function is degree 2.

Homothetic function

A homothetic function is a special type of homogeneous function where scaling of inputs by some factor, say $\lambda$ will scale the output by the same factor $\lambda$ without affecting the shape of the function.

A function is homothetic if it is a monotonic transformation of a homogenous function. In other words, a homogeneous function of degree 1 is a homothetic function.

A monotone transformation is a mathematical operation applied to a function that preserves the order of values. In simpler terms, if you have two inputs where one is greater than the other, the monotone transformation ensures that the transformed outputs maintain the same order.

 Formally, a function $T:R \rightarrow R$ is called monotone if $\forall$ $x_1$ and $x_2$ in its domain where $x_1 < x_2$, then the corresponding transformed values $T(x_1)$ and $T(x_2)$ also follow the inequility $T(x_1) < T(x_2)$ (for a strict monotonic transformation) or  $T(x_1) \leq T(x_2)$ (for a weak monotonic transformation).

Let's consider a monotone

$g(z) =\sqrt{z}$ 

$g(f(x,y))=\sqrt{xy}---(iii)$

Now, let's check if equation (iii) is homothetic by increasing x and y by $\lambda$ fold.

$g(f({\lambda}x,{\lambda}y))=\sqrt{{\lambda}x{\lambda}y}$

$g(f({\lambda}x,{\lambda}y))=\sqrt{{\lambda}^2xy}$

$g(f({\lambda}x,{\lambda}y))={\lambda}\sqrt{xy}$

$g(f({\lambda}x,{\lambda}y))={\lambda}g(f(x,y))---(iv)$

Equation (iv) is a homogeneous function of degree 1, so Equation (iii) is homothetic function. Other explanation, if $x>y$, then $\sqrt{x}>\sqrt{y}$ and $f(x,y)>g(f(x,y))$. So, Equation (iii) is is a homothetic function.