Structural subtyping and parametric polymorphism provide similar
flexibility and reusability to programmers. For example, both features
enable the programmer to provide a wider record as an argument to a
function that expects a narrower one. However, the means by which they
do so differs substantially, and the precise details of the
relationship between them exists, at best, as folklore in literature.

In this paper, we systematically study the relative expressive power
of structural subtyping and parametric polymorphism. We focus our
investigation on establishing the extent to which parametric
polymorphism, in the form of row and presence polymorphism, can encode
structural subtyping for variant and record types. We base our study
on various Church-style $\lambda$-calculi extended with records and
variants, different forms of structural subtyping, and row and
presence polymorphism.

We characterise expressiveness by exhibiting compositional
translations between calculi. For each translation we prove a type
preservation and operational correspondence result. We also prove a
number of non-existence results. By imposing restrictions on both
source and target types, we reveal further subtleties in the
expressiveness landscape, the restrictions enabling otherwise
impossible translations to be defined. More specifically, we prove
that full subtyping cannot be encoded via polymorphism, but we show
that several restricted forms of subtyping can be encoded via
particular forms of polymorphism.