  Let $X$ be an irreducible smooth projective curve defined over
$\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two.
For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the
Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of
the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm
Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective
cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When
$r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki.
