I am an economist, and in a research paper I am working on, I came accross an harmonic mean. I am not very familiar with this kind of mean, and did my best to study the topic, but I have a few questions I am not sure how to think about (and depending on which I would be able to obtain nice economic intuitions - about how to tax incomes!).
Consider the "smooth" function $p\in[0,1] \rightarrow x(p)$, with $x(p)>0$, $x^\prime(p)<0$ and $x^{\prime\prime}(p)<0$.
Let $h(p)=\frac{1-p}{\int_p^1 \frac{1}{x(\pi)} d\pi}$. This is the harmonic mean of $x$ above $p$.
Because $x$ is decreasing, $h(p)$ decreases with $p$. Right?
I would like to:
- compare the rate of increase of $x(p)$ with the rate of increase of $1/h(p)$. More precisely, is $x(p)/h(p)$ increasing or decreasing in $p$ given the above assumptions?
- say something about whether $x (p) \left[\frac{1}{h(p)}-\frac{1}{h(0)}\right]$ is larger or lower than $1$.
Any help would be most appreciated. I am a little bit stuck right now...
Thanks!