In this article, we study a life-cycle model of health stock and health spending. We use a simplified version of the model Grossman (1972). Contrary to Laporte (2015) and Strulik (2015), we suppose that the depreciation rate of health stock increases with age. Lifespan is thus an endogenous variable. We also suppose that the multiplicative productivity term of the income function (depending on the health of the agent) displays a hump shaped profile over time. Our contribution is to investigate the Grossman (1972)'s model in a non stationary environment with an endogenous finite lifetime. After derivation of the optimality conditions the model is solved numerically and we simulate the path followed by the main variables over the agent's life-cycle. Calibrated on the US economy with realistic values of parameters, the model is able to replicate a growing profile of health spending and a significant share of health spending at the end of life roughly in accordance with the data.