Social stress can cause physical and mental harm. It is therefore not surprising that public health policy makers have sought to identify and implement policies aimed at tackling this social ill. A frequently prescribed remedy is to reduce social stress by reducing income inequality, which is typically measured by the Gini coefficient. Decomposing the coefficient into a measure of a population's social stress and a population's income makes it possible to show that steps taken to lower the coefficient can actually exacerbate social stress. We formulate conditions under which lowering the Gini coefficient coincides with increasing social stress. If the aim of public policy is to improve public health and increase social welfare, and if social welfare is reduced by social stress, then lowering the Gini coefficient may not be the right course of action.