A main construction principle for bivariate copulas with desirable tail properties uses the multiplicative representation $R(W_1, W_2)$, where $R$ is a univariate scaling variable, and $W = (W_1, W_2)$ is a bivariate random vector. Numerous models in extreme value statistics are particular cases of this construction, and, depending on the distributions of $R$ and $W$, they can result in either asymptotic dependence or asymptotic independence. We systematically characterize the extremal dependence structures arising from such multiplicative constructions. It turns out to be crucial how the tail decay rate in $R$ impacts the tail dependence of $(W_1, W_2)$. The results allow to recover the extremal properties of existing models in a unified way, and, on the other hand, they can be used to construct new statistical models with flexible tail (in)dependence structures. The theory can also be applied to understand tail properties of spatial models. This is joint work with Thomas Opitz and Jennifer Wadsworth. ​