Consider the function $$ f(x,y) = g(x) y $$ where $g$ is some other function. We can restrict ourselves to $y\geq 0$ and $0\leq x\leq 1$. I would like to find a function $g$ with the following properties:
- $f$ is concave
- $g(x)\geq 0 $ for $x\in [0,1]$
- $g$ looks like a rough inverted U, i.e. there exists an $0<x^*<1$ such that $g$ is increasing before $x^*$ and decreasing after it.
Does such a function $g$ exist? If so it would be great to have an example. Many thanks!
The Hessian of $f$ is: $$\begin{pmatrix}g''(x)y & g'(x) \\ g'(x) & 0\end{pmatrix}$$ The eigenvalues need to be nonpositive. Since the trace is the sum of the eigenvalues and the determinant is the product of the eigenvalues, you need $g''(x)y \leq 0$ and $-(g'(x))^2 \geq 0$. The last condition implies that $g$ is constant.