  The theory of ordinal ranks on Baire class 1 functions developed by Kechris
and Loveau was recently extended by Elekes, Kiss and Vidny\'{a}nszky to Baire
class $\xi$ functions for any countable ordinal $\xi\geq1$. In this paper, we
answer two of the questions raised by them in their paper (Ranks on the Baire
class $\xi$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143).
Specifically, we show that for any countable ordinal $\xi\geq1,$ the ranks
$\beta_{\xi}^{\ast}$ and $\gamma_{\xi}^{\ast}$ are essentially equivalent, and
that neither of them is essentially multiplicative. Since the rank $\beta$ is
not essentially multiplicative, we investigate further the behavior of this
rank with respect to products. We characterize the functions $f$ so that
$\beta(fg)\leq \omega^{\xi}$ whenever $\beta(g)\leq\omega^{\xi}$ for any
countable ordinal $\xi.$
