I have the following (possibly quick) question. In a paper I am working with, the following conclusions are drawn which I have a hard time to understand. Since they are given without proof, I assume that my mind has not woken up properly yet and I will greatly appreciate help!
First conclusion: Let U be a strictly concave function on $\mathbb{R}^+$ and let $w,z,y>x\geq 0$. Then, the equality $U(y)-U(x) = U(w) - U(z)$ implies due to strict concavity of $U$ that $w-z > y-x$.
- Graphically, I agree with the conclusion but I'd like to have a technical proof.
Second conclusion (converse): For a function $U$ and for all $w,z,y>x\geq 0$ such that $w-z > y-x$ it holds that $U(y)-U(x) = U(w) - U(z)$. Hence, $U$ is concave on $\mathbb{R}^+$.
- In the paper they just write concave. Do they mean strictly concave?
- Again, can anyone show me where the conclusion comes from?
I can see that both conclusions arise from an if and only if statement about (strictly) concave functions, however for some reason I am not able to derive this statement.
If someone knows of a result for convex functions that would obviously help as well.
Thank you for any input, I am just stuck at the moment :(.
Best, Martin