The social cost of carbon (SCC) has been the central economic measure for climate damages, as recently highlighted by the US Interagency Group deriving SCC estimates for US regulatory impact assessments. The SCC measures the additional damage caused by an extra unit of emissions. Its level is therefore critically influenced by preferences for equity if damages fall on poor households. Yet, most SCC estimates are derived from models that neglect inequality at the household level. Our study accounts for household inequality and derives the SCC as the optimal carbon price in a two-level governance model. We introduce a previously neglected information asymmetry between the two levels. The global level knows the distribution of household income, costs and benefits of climate policy and internalizes climate damages according to a social welfare function. However, only the national governance level knows the identity of households and redistributes between them according to its own preferences. We show analytically that the SCC depends on national redistribution. If climate damages fall over-proportionally on poor households in one country and are not compensated, the SCC increases for every country compared to the case in which all households are equal. Numerical modeling shows that the increase in the SCC is large if damages of poor households are large. On the other hand we show that if national redistribution is based on a national social welfare function, the SCC changes only moderately when introducing household inequality. Our results are important for analyzing optimal carbon taxes in aggregated numerical models.