Consider the multivariate real-valued function
$$ y = f(x_1,...,x_n) $$
with $y, x_i \in R^+ \,\forall i$.
What is the class/type of functions $f(\cdot)$ that exhibit the following property?
$$ \frac{\partial y}{\partial x_i} = c_i \left(\frac{y}{x_i}\right)^{a_i} \quad \forall i $$
where $a_i, c_i$ are constants.
This is, the partial derivative is a proportional function of the ratio of the output over the input.
Examples of $f(\cdot)$ include:
- monomials of any homogeneity degree (including one-dimensional power functions). For example:
$$ y = x_1^{0.3}x_2^{0.6}x_3^{0.5} $$
- CES homogeneous of degree 1. For example:
$$ y = \left(0.3 x_1^\rho + 0.4 x_2^\rho + 0.3 x_3^\rho\right)^{\frac{1}{\rho}} $$
Notice monomials are a special case of a CES function. Also, notice that homogeneity of degree 1 is not sufficient for the property to hold. For instance, $y = x_1 + x_2$ does not fulfill the above condition.
Is there another necessary, "primitive" condition(s) on $f(\cdot)$ that fully characterise such class of functions? Does this class of functions have a name?
I thought I could find the general form of $f(\cdot)$, perhaps using the property from homogeneity of degree $k$:
$$ k y = \sum^n_i x_i \frac{\partial y}{x_i}$$
But combining the two (replacing the first one into the second one) seem to lead to nowhere. Maybe you can think of another way?
I think I got it, at least for the special case of $a_i=a$.
Replacing
$$ \frac{\partial y}{\partial x} = c_i \left(\frac{y}{x_i}\right)^{a} $$
into (assuming homogeneous degree 1)
$$ y = \sum x_ \frac{\partial y}{\partial x_i} $$
you get:
$$ y = \sum x_ c_i \left(\frac{y}{x_i}\right)^{a} $$
Solving for $y$ gives:
$$ y = \left(\sum c_i x_i^{1-a}\right)^{\frac{1}{1-a}} $$
which is the exact definition of a CES with homogeneous degree 1.
Furthermore, consider the special case of $a_i=a=1$, without assuming homogeneous degree 1. This is the case when the CES becomes a monomial (known in economics as a Cobb-Douglas function). Here we have
$$ ky = \sum x_ c_i \left(\frac{y}{x_i}\right) $$
from where we find that
$$ k = \sum c_i $$
So, the property holds for every $y$ of that nature, which the added result that the homogeneity degree is given by the sum of the exponents, a well known result.
I don't know how to proceed with the general case of heterogeneous $a_i$. Perhaps one could rule out the existence of a function $y$ in that given case. I don;t know how to do this, and for now I will leave it as that.
Finally, one issue was puzzling me, which is the odd asymmetry between the CES case with homogeneous degree 1 and the not-quite nested case of monomial with any homogeneity degree. My experience in mathematics tells me something smells fishy here. And I think I figured it out.
In particular, as obvious as it seems, the property I was looking at is not general enough to cover CES with homogeneity degree different to 1. So, let's take the CES in the example of the question, but with degree of homogeneity $k$:
$$ y=(0.3_1^1+0.4_2^+0.3_3^)^{\frac{k}{}} $$
Then
$$ \frac{\partial y}{\partial x_1} = 0.3k y^{1-\frac{\rho}{k}} x_1^{\rho-1} $$
So, the actual formula I was interested is actually
$$ \frac{\partial y}{\partial x} = kc_i \frac{y^{1-\frac{1-a_i}{k}}}{x^{a_i}} $$
which reduces to the one in the question in the special case of $k=1$.
it is trivial to "show" the above property allows for a CES with any homogeneity degree. Just proceed as before (assuming $a_i=a$), and you get:
$$ ky = \sum k x_ c_i \frac{y^{1-\frac{1-a}{k}}}{x^{a}} $$
Solving for $y$ yields:
$$ y = \left(\sum c_i x^{1-a}\right)^{\frac{k}{1-a}} $$
which is the general CES formula.
PS: in case you are wondering what's all have to do with anything, I am trying to solve this question, and I think this property is a key part of the solution. I'm still working on it.