To study the trade-offs and the macroeconomic repercussions of rising health care demand in a public health service, we develop a continuous-time overlapping generations model with a public health care sector and a realistic aging process. Health care services are provided to two groups of individuals, the healthy and the sick, free of charge at the point of service. Without a price mechanism, the government relies on a queuing rule for allocating its services. We conceptualize this mechanism as congestion that lowers the efficacy of health care. Then, we calibrate the model to match UK data from 2007–2016 and analyze the steady-state, general equilibrium response of the economy of shocks to productivity/income and medical effectiveness. Our analysis suggests that the optimal response to an increase in the demand for health care depends strongly on whether it is due to an increase in income or medical effectiveness. We also show that there is disagreement across age-groups on the preferred policy.