Summary: We analyze a simple stochastic model of economic growth in which physical and health capital accumulation jointly contribute to determine long run economic growth. Health capital is subject to random shocks via the effects of behavioral changes: unpredictable changes in individuals’ attitude toward healthy behaviors may reduce the effectiveness of health services provision; this in turn, by reducing the production of new health capital, lowers economic production activities negatively affecting economic growth. Unlike the extant literature, we assume that the probability with which such random shocks occur is not constant but state-dependent. Specifically, the probability that behavioral changes will negatively impact on health capital and economic growth depends on the level of economic development, proxied by the relative abundance of health capital with respect to physical capital. We show that our model’s dynamics can be converted into an iterated function system with state-dependent probabilities which converges to an invariant self-similar measure supported on a (possibly fractal) compact attractor. We develop a numerical method to approximate the invariant distribution to illustrate its features in specific model’s parametrizations, exemplifying thus the effects of state-dependent probabilities on the model’s steady state.