UPDATE: added formulas of my attempts to prove results.
I am hitting some patterns that must have some sort of explanation. I can't find what it might be.
Consider the following optimisation problem:
\begin{equation*} \begin{aligned} & \underset{x,w,z \,\in \, R^+_0}{\text{maximize}} & & y(x,w,z) - m(x,w,z) \\ & \text{subject to} & & y(x,w,z) = x^{a_x} w^{a_w} z^{a_z},\\ & & & m(x,w,z) = xc_x + wc_w + zc_z,\\ &&& a_i,c_i \quad\text{are positive constants}\end{aligned} \end{equation*}
Nothing special. Just a trivial problem. The key is that $y$ is an homogeneous function.
Rather than solving the problem, let's derive the "conditional" function of $z$ in terms of $x$ and $w$. We know the three are chosen simultaneously, but we can "reduce" the dimensionality of the problem by reducing one of the variables to the others.
So, consider $z$. The first order condition for it gives $\dfrac{\partial y}{\partial z} = c_z$. From here we get:
$$ \hat z(x,w) = \left(\frac{a_z}{c_z}x^{a_x} w^{a_w}\right)^{\frac{1}{1-a_z}}$$
Now, let us replace $\hat z(x,w)$ into $y$, so that we end up with a two-dimension problem. The new function $\hat y(x,w)$ is:
$$ \hat y(x,w) = \left(\frac{a_z}{c_z}\right)^{\frac{a_z}{1-a_z}} \left(x^{a_x} w^{a_w}\right)^{\frac{1}{1-a_z}} $$
So, here are the curious results (to me):
the conditional functions $\hat z(x,w)$ and $\hat y(x,w)$ have the same functional form. They are just a scale of each other. Same homogeneity degree.
the conditional function $\hat y(x,w)$ has the "same" functional form than the unconditional one. Naturally, when we assume $a_z=0$, they are exactly the same. This is, they nest each other. Their homogeneous degree is not the same, but they are both homogeneous anyway.
This is, it seems $\hat y(x,w) = A \hat z(x,w)$ and, if you forbid my misuse of notation, $\hat y(\cdot) = y(\cdot)$.
I have confirmed the above for other homogeneous functions too, like CES. It is trivial to check that the above does not hold when $y(\cdot)$ is not homogeneous, for example, if $y(x,w,z) = x^{a_x} + w^{a_w}z^{a_z}$.
I'm pretty sure these results, particularly point 1, have a name. I have failed so far. Tried checking envelope theorem, implicit function stuff, Roy's identity, but failed (not that I am a great mathematician to be honest). Can you guide me on this please?
UPDATE:
as said above, I've tried different avenues to prove the above results. Let's focus on point 1. I want to prove that $\hat y(x,w) = A \hat z(x,w)$, with $A$ a constant.
The target function $y(x,w,z)$ is homogeneous. It follows that any partial derivative from it also is. Therefore, so is $\dfrac{\partial y}{\partial z}$. Since the FOC is $\dfrac{\partial y}{\partial z} = c_z$, we can define the implicit function
$$ h(x,w,z) - c_z = 0$$
where $h(\cdot)=\dfrac{\partial y}{\partial z}$, also homgeneous. From here we know that:
$$ \frac{dz}{dx} = - \frac{\partial h/\partial x}{\partial h/\partial z} $$
$$ \frac{dz}{dw} = - \frac{\partial h/\partial w}{\partial h/\partial z} $$
Not sure how to proceed for this avenue.
Another way to tackle the issue is to realize $\hat z(x,w)$ is also homgeneous with given degree $k$ (although not sure how to prove this). Then, by properties of homogeneous functions it follows that:
$$ k \hat z(x,w) = x\frac{\partial \hat z(x,w)}{\partial x} + w\frac{\partial \hat z(x,w)}{\partial w} $$
But again, I'm lost here.