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This paper introduces the concept of unintentional bequests in a closed economy à la Chakraborty (J Econ Theory 116:119-137, ) with overlapping generations. We show that scarce public investments in health can lead to poverty traps depending on the relative size of the output elasticity of capital. More importantly, the existence of unintentional bequests, rather than a market for annuities, means that health tax rates play a prominent role in determining the stability of the long-term equilibrium in rich economies. In fact, Neimark-Sacker bifurcations and endogenous fluctuations occur depending on the size of the public health system. [ABSTRACT FROM AUTHOR]