Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between R and RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. Our first application concerns the asymptotic behaviour of the componentwise sample minima related to elliptical distributions. Further, we derive the lower tail asymptotics of the aggregated risk for bivariate polar distributions.