Productivity of maximal eventually different families

A maximal eventually different family is called $n$-productive if the product family ${\mathcal{F}}^n$ is still maximal. We construct closed $n$-productive families separating these strengthenings of maximality at every $n\ge 1$. Furthermore, we show how to force and construct an even stronger type of ${\mathcal{I}}_0$-productive family and discuss the relation of productivity to Van Douwen families.