Reducing social inequality and protecting the environment are two distinct objectives that can both complement and contradict each other. In this study, we examine the impact of climate risks on social inequality and demonstrate a potential trade-off between environmental and social improvements. The first part is dedicated to the incorporation of the social risk (intergenerational, interregional and intraregional inequalities) into climate economic modeling. We notice that taking social inequalities into account changes substantially the optimal price of the carbon tax and its efficiency. In a second part, we empirically study social inequality in the context of the transition risk in France. We show that the effort rate of the poorest 10% of households is 2.8 times higher than the richest 10% for a 100 euros per ton of CO2e carbon tax. If the regressive nature of the tax can be mitigated by redistribution policies, compensation may induce some potential backfire effects. It follows that implementing a carbon tax is challenging and its theoretical benefits may be lower in practice.