Does concavity imply that $f'(a)<\frac {f(a)}a$?

Question

Does concavity imply that $f'(a)<\frac {f(a)}a$ ? I have read this in the proof of a paper by Straub, Mian and Sufi (https://scholar.harvard.edu/files/straub/files/mss_indebteddemand.pdf), appendix B.1. and I did not know this before. Does anybody know the proof of this or why this holds?

Answer 1

See equation (3) page 7, $\eta(a)=a\nu'(a)$. Thus if $\nu$ is concave, $\nu''<0$ so, $$\eta'(a)=\nu'(a)+a\nu''(a)<\nu'(a)=\frac1a(a\nu'(a))=\frac{\eta(a)}{a}.$$