  The order and chain polytopes are two 0/1-polytopes constructed from a finite
poset. In this paper, we study the $f$-vectors of these polytopes. We
investigate how the order and chain polytopes behave under disjoint unions and
ordinal sums of posets, and how the $f$-vectors of these polytopes are
expressed in terms of $f$-vectors of smaller polytopes. Our focus is on
comparing the $f$-vectors of the order and chain polytope built from the same
poset. In our main theorem we prove that for a family of posets built
inductively by taking disjoint unions and ordinal sums of posets, for any poset
$\mathcal{P}$ in this family the $f$-vector of the order polytope of
$\mathcal{P}$ is component-wise at most the $f$-vector of the chain polytope of
$\mathcal{P}$.
