Positively Homogenous Functions of Several Variables

Positively Homogenous Functions of Several Variables

 Definition: Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function. $f$ is said to be Positively Homogenous of Degree $k$ if for all $(x_1, x_2, ..., x_n) \in D(f)$ and for every real number $t > 0$ we have that $f(tx_1, tx_2, ..., tx_n) = t^k f(x_1, x_2, ..., x_n)$.

Let's look at some examples of homogenous functions. Let $f(x, y) = \frac{3x^2y^2}{\sqrt{x^4 + y^4}}$, and let $t > 0$. Then we have that:

(1)
\begin{align} \quad \quad f(tx, ty) = \frac{3(tx)^2(ty)^2}{\sqrt{(tx)^4 + (tx)^4}} = \frac{3t^2x^2t^2y^2}{\sqrt{t^4x^4+t^4x^4}} = \frac{t^4(3x^2y^2)}{\sqrt{t^4(x^4 + y^4)}} = \frac{t^4(3x^2y^2)}{t^2 \sqrt{x^4 + y^4}} = t^2 f(x, y) \end{align}

Therefore $f$ is positively homogenous of degree $2$. The graph of $f$ is shown below. Let's look at another example. Let $g(x, y, z) = xyz + x^3 + y^3 + z^3$ and let $t > 0$. Then we have that:

(2)
\begin{align} \quad \quad g(tx, ty, tz) = txtytz + (tx)^3 + (ty)^3 + (tz)^3 = t^3xyz + t^3x^3 + t^3y^3 + t^3z^3 = t^3 g(x,y,z) \end{align}

Therefore $g$ is positively homogenous of degree $3$. Of course, a positively homogenous function need not be of positive degree, i.e., the degree of a positively homogenous function can be nonpositive. For example, let $h(x, y) = \frac{x + y}{x^3y^3}$, and let $t > 0$. Then:

(3)
\begin{align} \quad \quad h(tx, ty) = \frac{tx + ty}{(tx)^3(ty)^3} = \frac{t(x + y)}{t^6(x^3y^3)} = t^{-5} h(x, y) \end{align}

Therefore $h$ is positively homogenous of degree $-5$. The graph of $h$ is shown below. One important property of positively homogenous functions of degree $k$ of two variables is that they have characteristics of their degree. For the function $f$ above, observe parabola shapes in the graph of $f$. Similarly, for the function $h$ above, observe the hyperbola shapes present. If the degree of a positively homogenous function is known, then this can considerably help us in graphing these two variable functions.

Euler's Positive Homogenous Function Theorem

 Theorem 1 (Euler's Positive Homogenous Function Theorem): Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function that is positively homogenous of degree $k$, and suppose that $f$ has continuous first partial derivatives. Then $\sum_{i=1}^n x_i \frac{\partial z}{\partial x_i} = k f(x_1, x_2, ..., x_n)$.
• Proof: Since $z = f(x_1, x_2, ..., x_n)$ is a positively homogenous function of degree $k$ then we have that:
(4)
\begin{equation} f(tx_1, tx_2, ..., tx_n) = t^k f(x_1, x_2, ..., x_n) \end{equation}
(5)
\begin{align} \quad \quad x_1 \frac{\partial}{\partial x_1} (tx_1, tx_2, ..., tx_n) + x_2 \frac{\partial}{\partial x_2} (tx_1, tx_2, ..., tx_n) + ... + x_n \frac{\partial}{\partial x_n} (tx_1, tx_2, ..., tx_n) = kt^{k-1} f(x_1, x_2, ..., x_n) \end{align}
• Since $f$ is positively homogenous for all $t > 0$, then by letting $t = 1$ we get that:
(6)
\begin{align} \quad \quad x_1 \frac{\partial }{\partial x_1} (x_1, x_2, ..., x_n) + x_2 \frac{\partial }{\partial x_2} (x_1, x_2, ..., x_n) + ... + x_n \frac{\partial }{\partial x_n} (x_1, x_2, ..., x_n) = k f(x_1, x_2, ..., x_n) \\ \quad \quad x_1 \frac{\partial z}{\partial x_1} + x_2 \frac{\partial z}{\partial x_2} + ... + x_n \frac{\partial z}{\partial x_n} = k f(x_1, x_2, ..., x_n) \\ \quad \quad \sum_{i=1}^{n} x_i \frac{\partial z}{\partial x_i} = k f(x_1, x_2, ..., x_n) \quad \blacksquare \end{align}