  Motivated by research on gender identity norms and the distribution of the
woman's share in a couple's total labor income, we consider functional additive
regression models for probability density functions as responses with scalar
covariates. To preserve nonnegativity and integration to one under vector space
operations, we formulate the model for densities in a Bayes Hilbert space,
which allows to not only consider continuous densities, but also, e.g.,
discrete or mixed densities. Mixed ones occur in our application, as the
woman's income share is a continuous variable having discrete point masses at
zero and one for single-earner couples. Estimation is based on a gradient
boosting algorithm, allowing for potentially numerous flexible covariate
effects and model selection. We develop properties of Bayes Hilbert spaces
related to subcompositional coherence, yielding (odds-ratio) interpretation of
effect functions and simplified estimation for mixed densities via an
orthogonal decomposition. Applying our approach to data from the German
Socio-Economic Panel Study (SOEP) shows a more symmetric distribution in East
German than in West German couples after reunification and a smaller child
penalty comparing couples with and without minor children. These West-East
differences become smaller, but are persistent over time.
