Derivation of Euler's theorem on homogeneous functions
1 Definition of homogeneous functions
A function \(f\) of variable \(x\) is said to be homogeneous of degree \(m\) if it satisfies the following equality:
\[\begin{equation} f\left(\lambda x\right) = \lambda^m \times f\left(x\right) \tag{1.1} \end{equation}\]
As an example, the function \(f(x)=x\) is homogeneous of degree \(1\) because \(f(\lambda x) = \lambda x\).
This definition can be extended to multi-variable functions. Let us consider a function \(f\) of variables \(x\) and \(y\). Then \(f\) is said to be homogeneous of degree \(m\) with respect to variable \(x\) if it satisfies the following equality:
\[\begin{equation} f\left(\lambda x, y\right) = \lambda^m \times f\left(x, y\right) \tag{1.2} \end{equation}\]
For example, the function \(f(x, y)=xy\) is homogeneous of degree \(1\) with respect to variable \(x\) because \(f(\lambda x, y) = \lambda xy\). One could also say that the function \(f(x, y) = xy\) is homogeneous of degree \(1\) with respect to variable \(y\) because \(f(x, \lambda y) = \lambda xy\). Finally, one could also say that the function \(f(x, y)=xy\) is homogeneous of degree \(2\) with respect to variables \(x\) and \(y\) because \(f(\lambda x, \lambda y) = \lambda^2 xy\). Thus, it is important to specify both (i) the homogeneous degree and (ii) the considered variables.
The variables which can be multiplied by a factor \(\lambda\) and that leads to a homogeneous function are called scalable parameters (because multiplying them by \(\lambda\) leads to increase the function value). Other variables are said to be non-scalable variables. As we will see soon, Euler’s theorem application depends on the considered scalable variables.
Finally, for functions of many variables, it is convenient to define vectors as variables: first variable/vector is called \(A\) and is filled with all scalable variables while the second vector/variable is called \(a\) and is filled with all non-scalable variables.
Let us consider the function \(f(x,y,z) = xy \times \exp\left(z\right)\). Because \(f(\lambda x,\lambda y,z) = \lambda^2 f(x,y,z)\), the scalable variables are \(x\) and \(y\) while the non-scalable parameter is \(z\). Thus, in a simpler way, the \(f\) function could write \(f\left(A,a\right)\) with \(A=(x,y)\) and \(a=z\). The homogeneity condition writes:
\[\begin{equation} f\left(a,\lambda A\right) = \lambda^m \times f\left(a, A\right) \tag{1.3} \end{equation}\]
2 Derivation of Euler’s theorem
Let us consider a homogeneous function of degree \(m\) of variables \(A\). Therefore, this function satisfies equation (1.3). Euler’s theorem states that function \(f\) satisfies the following equation:
\[\begin{equation} f\left(a, \lambda A\right) = \lambda^m \times f\left(a, A\right) \Leftrightarrow \sum_i \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} \times A_i = m \times f\left(a, A\right) \tag{2.1} \end{equation}\]
2.1 Left to right implication
To obtain this equality, let us write the differential of equation (1.3).
- First, let define the function \(g\left(\lambda, A\right) = \lambda \times A\). Differential of LHS then writes:
\[\begin{equation} df\left(a,g\right) = \sum_i \left(\frac{\partial f}{\partial a_i}\right)_{a_{j \neq i}, g} da_i + \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} dg_i \tag{2.2} \end{equation}\]
The differential element \(dg_i\) can be written with respect to variables \(\lambda\) and \(A_i\):
\[\begin{equation} dg_i = \lambda dA_i + A_i d\lambda \tag{2.3} \end{equation}\]
Thus, equation (2.2) writes:
\[\begin{equation} df\left(a,A, \lambda\right) = \sum_i \left(\frac{\partial f}{\partial a_i}\right)_{a_{j \neq i}, g} da_i + \lambda \times \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} dA_i + \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} A_i d\lambda \tag{2.4} \end{equation}\]
- Differential of the RHS writes:
\[\begin{equation} d\left[\lambda^m \times f\left(a, A\right)\right] = m \times \lambda^{m-1} f\left(a, A\right) \times d\lambda + \lambda^m \left[\sum_i \left(\frac{\partial f}{\partial a_i}\right)_{a_{j \neq i}, A} da_i + \sum_i \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} dA_i\right] \tag{2.5} \end{equation}\]
By identifying coefficients before \(da_i\), \(dA_i\) and \(d\lambda\) elements, one can write:
\[\begin{equation} \begin{cases} \left(\frac{\partial f}{\partial a_i}\right)_{a_{j \neq i}, g} = \lambda^m \times \left(\frac{\partial f}{\partial a_i}\right)_{a_{j \neq i}, A} \\ \lambda \times \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} = \lambda^m \times \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} \\ \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} A_i = m \times \lambda^{m-1} f\left(a, A\right) \end{cases} \tag{2.6} \end{equation}\]
Second equality of (2.6) rewrites:
\[\begin{equation} \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}} = \lambda^{m-1} \times \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} \tag{2.7} \end{equation}\]
Injecting this expression of \(\left(\frac{\partial f}{\partial g_i}\right)_{a,g_{j \neq i}}\) in the third equality of (2.6), one obtains:
\[\begin{equation} \lambda^{m-1} \sum_i \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} A_i = m \times \lambda^{m-1} f\left(a, A\right) \tag{2.8} \end{equation}\]
Dividing equation (2.8) by \(\lambda^{m-1}\), the Euler theorem is obtained:
\[\begin{equation} \sum_i \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} A_i = m \times f\left(a, A\right) \tag{2.9} \end{equation}\]
2.2 Right to left implication
So far, one has only demonstrated that a homogeneous function leads to Euler’s theorem. However, one might be interested to derive the opposite, i.e. that a function is homogeneous if it satisfies Euler’s theorem. First, we note that equation (2.1) allows us to write:
\[\begin{equation} \sum_i \left(\frac{\partial f}{\partial \lambda A_i}\right)_{a, \lambda A_{j \neq i}}\left(a, \lambda A\right) \times \lambda A_i = m \times f\left(a, \lambda A\right) \tag{2.10} \end{equation}\]
Setting \(g = \lambda \times A\), one gets:
\[\begin{equation} \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}}\left(a, g\right) \times g_i = m \times f\left(a, g\right) \tag{2.11} \end{equation}\]
From (2.4), one gets:
\[\begin{equation} \left(\frac{\partial f}{\partial \lambda}\right)_{a, A} = \sum_i \left(\frac{\partial f}{\partial g_i}\right)_{a, g_{j \neq i}}\left(a, g\right) \times A_i \tag{2.12} \end{equation}\]
Combining equations (2.11) and (2.12), one finds that:
\[\begin{equation} \lambda \times \left(\frac{\partial f}{\partial \lambda}\right)_{a, A} = m \times f\left(a, \lambda A\right) \tag{2.13} \end{equation}\]
If one separates the variables and performs integration:
\[\begin{equation} \int_{f\left(a, A\right)}^{f\left(a, \lambda A\right)} \frac{df}{f} = m \times \int_{1}^{\lambda} \frac{d \lambda}{\lambda} \tag{2.14} \end{equation}\]
After integration:
\[\begin{equation} \ln\left[\frac{f\left(a, \lambda A\right)}{f\left(a, A\right)}\right] = m \times \ln\left(\frac{\lambda}{1}\right) \tag{2.15} \end{equation}\]
Finally, one obtains that the \(f\) function satisfies the homogeneity condition:
\[\begin{equation} f\left(a, \lambda A\right) = \lambda^m \times f\left(a, A\right) \tag{2.16} \end{equation}\]
3 Higher degree Euler theorem
As seen in equation (2.1), Euler’s theorem is primarily intended to give a mathematical relation between the first derivatives of the \(f\) function with respect to its scalable variables and the \(f\) function. As we will see, it is possible to derive a mathematical constraint involving the \(f\) function higher derivatives.
Let us start by deriving equation (2.1) with respect to variable \(A_j\):
\[\begin{equation} \sum_{i \neq j} \left(\frac{\partial^2 f}{\partial A_j \partial A_i}\right)_{a, A_{j \neq i}} A_i + \left(\frac{\partial^2 f}{\partial A_i^2}\right)_{a, A_{j \neq i}} A_i + \left(\frac{\partial f}{\partial A_i}\right)_{a, A_{j \neq i}} = m \times \left(\frac{\partial f}{\partial A_i}\right) _{a, A_{j \neq i}} \tag{3.1} \end{equation}\]
After some rearrangements, one obtains:
\[\begin{equation} \sum_{i} \left(\frac{\partial^2 f}{\partial A_j \partial A_i}\right)_{a, A_{j \neq i}} A_i = \left(m-1\right) \times \left(\frac{\partial f}{\partial A_j}\right) _{a, A_{i \neq j}} \tag{3.2} \end{equation}\]
Because of Euler’s theorem, the functions \(\left(\frac{\partial f}{\partial A_j}\right)_{a, A_{i \neq j}}\) are homogeneous of degree \(m-1\).