This article aims at studying a general equilibrium model with overlapping generations that incorporates inherited tastes (aspirations) and endogenous longevity. The existence of standard-of-living aspirations transmitted between two subsequent generations in a context where the individual state of health depends on public investments in health has some remarkable consequences at the macroeconomic level. First, aspirations allow escaping from the well-known poverty trap scenario described by Chakraborty (2004). Second, the steady-state equilibrium may be destabilized through a super-critical Neimark–Sacker bifurcation when the health tax rate is set at too high or too low a level. This causes endogenous fluctuations in income and longevity.