We investigate whether people enrolled into voluntary health insurance (VHI) substitute public consumption with private (opt out) or just enlarge their private consumption, without reducing reliance upon public provisions (top up). We study the case of Italy, where a mixed insurance system is in place. To this purpose, we specify a joint model for public and private specialist visits counts, and allow for different degrees of endogenous supplementary insurance coverage, looking at the insurance coverage as driven by a trinomial choice process. We disentangle the effect of income and wealth by going through two channels: the direct impact on the demand for healthcare and that due to selection into VHI. We find evidence of opting out: richer and wealthier individuals consume more private services and concomitantly reduce those services publicly provided through selection into for-profit VHI. These results imply that the market for VHI eases the redistribution from high income (doubly insured) individuals to low income (not doubly insured) ones operated by the Italian National Health Service (NHS). Accounting for VHI endogeneity in the joint model of the two counts is crucial to this conclusion.