  Let $(Z_n)$ be a supercritical branching process in an independent and
identically distributed random environment $\xi$. We show the exact decay rate
of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j
\geq k,$ assuming that $\mathbb{P} (Z_1 = 0) =0$. We also determine the
critical value for the existence of harmonic moments of the random variable
$W=\lim_{n\to\infty}\frac{Z_n}{\mathbb E (Z_n|\xi)}$ under a simple moment
condition.
