The economic impacts of climate change are highly uncertain. Two of the most important uncertainties are the sensitivity of the climate system and the so-called damage functions, which relate climate change to economic costs and benefits. Despite broad awareness of these uncertainties, it is unclear which of them is most important, especially at the regional level. Here we construct regional damage functions, based on two different global damage functions, and apply them to two climate models with vastly different climate sensitivities. We find that uncertainty in both climate sensitivity and aggregate economic damages per degree of warming are of similar importance for the global economic impact of climate change, with the decrease in global economic productivity ranging between 4% and 24% by the end of the century under a high-emission scenario. At the regional level, however, the effects of climate change can vary even more substantially, depending both on a region's initial temperature and the amount of warming it experiences, with some regions gaining in productivity and others losing. The ranges of uncertainty are therefore potentially much larger at a regional level. For example, at the end of the century, under a high-emission scenario, we find that India's productivity decreases between 13% and 57% and Russia's increases between 24% and 74%, while Germany's change in productivity ranges from an increase of 8% to a decrease of 4%. Our findings emphasize the importance of including these uncertainties in estimates of future economic impacts, as they are vital for the resulting impacts and thus policy implications.