Equity considerations play an important role in international climate negotiations. While policy analysis has often focused on equity as it relates to mitigation costs, there are large regional differences in adaptation costs and the level of residual damage. This paper illustrates the relevance of including adaptation and residual damage in equity considerations by determining how the allocation of emission allowances would change to counteract regional differences in total climate costs, defined as the costs of mitigation, adaptation, and residual damage. We compare emission levels resulting from a global carbon tax with two allocations of emission allowances under a global cap-and-trade system: one equating mitigation costs and one equating total climate costs as share of GDP. To account for uncertainties in both mitigation and adaptation, we use a model-comparison approach employing two alternative modeling frameworks with different damage, adaptation cost, and mitigation cost estimates, and look at two different climate goals. Despite the identified model uncertainties, we derive unambiguous results on the change in emission allowance allocation that could lessen the unequal distribution of adaptation costs and residual damages through the financial transfers associated with emission trading.