Dtjytyk

In short, my question – which has come up in a mechanism design setting I'm working on – is the following. Let $f,gcolon mathbb{R} rightarrow mathbb{R}$ be continuous functions and let $f$ be non-decreasing and non-negative. Then for each $yinmathbb{R}$ consider the function
$$
r_ycolon mathbb{R} rightarrow mathbb{R}colon x mapsto f(x)cdot y-g(x)
$$
over $mathbb{R}$. What do $f,g$ have to look like for $r_y$ to be globally optimal at $x=y$ for all $yin mathbb{R}$?

Of course, if $f,g$ are somewhat "well-behaved", we can go through the usual motions. For all $y$ it has to be $r_y'(y)=0$, i.e. $g'(y)=f'(y)y$ and via the fundamental theorem of calculus
$$
g(x)=C+int_0^x f'(zeta)zeta dzeta.
$$

But my calculus/real analysis knowledge is insufficient to understand whether the premises of the problem ensure that $f,g$ are sufficiently well-behaved or to understand what happens if they are not well-behaved.

Here are some more thoughts of mine. It's been a while since I took advanced calculus, so some of this might be confused.

• Instead of talking about derivatives, we can talk about the (ratios of) differences. In particular, for all $d>0,yin mathbb{R}$, it must be $r_y(y+d)-r_y(y)leq 0$ and $r_y(y)-r_y(y-d)geq 0$. Rearranging stuff a little, this is equivalent to (for all $d>0,yin mathbb{R}$)
  $$
  y(f(y+d)-f(y))leq g(y+d)-g(y)leq (y+d)(f(y+d)-f(y)).
  $$
  This is nice because it's similar to the relationship between the derivatives. (Dividing by $d$ and letting it go to $0$ gives the relationship between the derivatives if they exist.) But it doesn't immediately answer questions about what kinds of $f$ are allowed or how to construct the corresponding $g$. (See below for problematic examples.)




• Lebesgue's theorem for the differentiability of monotone functions seems to imply that $f$ and $g$ are differentiable almost everywhere. (The previous point implies that if $f$ is monotone, $g$ is monotone on $(-infty,0)$ and on $(0,infty)$.) So we can talk about the derivatives of $f$ and $g$ almost everywhere. But we might not be able to obtain (an "allowed") $g$ via integration of $g'(x)=f'(x)x$. For example, if $f$ is the Cantor function for $xin[0,1]$, then
  $$
  int_0^x f'(zeta)zeta dzeta = 0,
  $$
  right (see, e.g., sect. 2 here)? But if $f$ is the Cantor function and $g=0$, then $r_{frac{1}{2}}(frac{1}{2})<r_{frac{1}{2}}(1)$. That is, $r_{frac{1}{2}}$ does not have a maximum at $1/2$. So can $f$ be the Cantor function? That is, if $f$ is the Cantor function, is there a corresponding $g$ (s.t. $r_y$ has a maximum at $y$ for all $y$) and if so what is the corresponding $g$? More generally, can $f$ be a function that we can't find by integrating its derivative and if so, is there a way of finding the corresponding $g$?




• The above point asks whether $g$ might not be an integral over $g'$. Another question is whether $g'$ (or $f'$) must be integrable at all. Of course, there are continuous functions whose derivatives are not integrable, but the examples I am aware of aren't even monotone (not to mention our other requirements for $f,g$).




• One standard move that people use to replace derivatives is to use sub-/super-derivatives. But that seems to require the functions in question to be convex/concave. But neither $r_y$, nor $f$ or $g$ must be convex/concave everywhere.