Suppose I have a function $f(x,y)$ defined on $[0,1]^2$ that is equal to the value of the following constrained minimization problem:
$$f(x,y) = min_{a\in [0,1-x),\ b\in[0,x)}\ \left\{ h(x,a) + g(x,b) \right\}\ \text{subject to } a+b=y$$
Here is some relevant information about $h$ and $g$. First, $h(x,0)=g(x,0)$ for all $x$. Second, $\lim_{a\rightarrow 1-x} h(x,a)=\infty$ and $\lim_{b\rightarrow x}g(x,b)=\infty$ (this is why I write the feasible sets as partially open intervals).
I am interested in an analytical characterization of the partial derivatives $f_x(x,y)$ and $f_y(x,y)$ in terms of the partial derivatives of $h$ and $g$. My envelope theorem is super rusty, and the simplicity of both the objective and the constraint make me think there is a relatively easy solution here that I am missing.
Note this is not homework (I am not a student, but rather an old professor who has come to rely perhaps too heavily on numerical methods for his own good).
Here is my logic so far:
(i) The simplest case is if $x=0$, where $b=0$ and $a=y$ by construction. Obviously in this case $f(0,y)$ = $h(0,y)$ for all $y$. Then $f_x(0,y) = h_x(0,y)$ and $f_y(0,y) = h_a(0,y)$.
(ii) If $x>0$ but we have a corner solution (either $a=0$ or $b=0$), the answer is still pretty simple. If $a=0$ then $f_x(x,y)=g_x(x,y)$ and $f_y(x,y)=g_b(x,y)$. If $b=0$ then $f_x(x,y) = h_x(x,y)$ and $f_y(x,y) = h_a(x,y)$ as in (i).
(iii) For interior solutions, the first-order conditions imply $h_a(x,a^*(x,y))=g_b(x,b^*(x,y))$, where $a^*(x,y)$ and $b^*(x,y)$ are the argmins. I am thinking $f_y(x,y)=h_a(x,a^*(x,y))=g_b(x,b^*(x,y))$ in this case. At the margin, allocating a small increase in y to a or b should have the same effect on the objective.
[iv] The only thing I am left with is: what is $f_x(x,y)$ for interior solutions?
I think I have solved this, but it would be extremely useful to get some feedback to verify this solution.
We can get rid of the constraint $a+b=y$ and write the problem as $$f(x,y)=\min_{a\in(y-x,y]\cap[0,x)}\ \{h(x,a)+g(x,y-a)\}$$ Restricting attention to interior solutions where $0<a<y$, the first-order condition is $$h_a(x,a^*(x,y))=g_b(x,y-a^*(x,y))$$
We can write $f_x$ as $$f_x(x,y)=h_x(x,a^*(x,y))+h_a(x,a^*(x,y))a^*_x(x,y)+g_x(x,y-a^*(x,y))-g_b(x,y-a^*(x,y))a^*_x(x,y)$$ Using the FOC, this simplifies to $$f_x(x,y)=h_x(x,a^*(x,y))+g_x(x,y-a^*(x,y))$$ This is a bit comforting, as it appears to be the standard envelope condition result.
We can write $f_y$ as $$f_y(x,y)=h_a(x,a^*(x,y))a^*_y(x,y)+g_b(x,y-a^*(x,y))\left[1-a^*_y(x,y)\right]$$ Using the FOC again, we get $$f_y(x,y)=g_b(x,y-a^*(x,y))=h_a(x,a^*(x,y))$$ This verifies the answer I proposed in item (iii) of my question.
For corner solutions, I believe the answers proposed in (i) and (ii) of my question are correct.