Viability theory is the study of dynamical systems that asks what set of initial conditions will generate evolutions that obey the laws of motion of a system and some state constraints, for the length of the evolution. We apply viability theory to Judd's dynamic tax model [The welfare cost of factor taxation in a perfect-foresight model, Journal of Political Economy 95(4), 675–709 (1987)] to identify which economic states today are sustainable under only slightly constrained tax-rate adjustments in the future, when the dynamic budget constraint and the consumers' transversality condition at infinity are satisfied. We call the set of such states the economic viability kernel. In broad terms, knowledge of the viability kernel can tell the planner what economic objectives are achievable and assist in the choice of suitable controls to realize them. We observe that high consumption levels can only be sustained when capital is abundant and, unsurprisingly, that a very high consumption economy lies outside such kernels, at least for annual tax-adjustment levels limited by 20 percentage points. Furthermore, we notice that by and large the sizes of the kernel slices do not diminish as the tax rate rises; hence high-taxation economies are not necessarily more prone to explode, or implode, than their low-taxation counterparts. In fact, higher tax rates are necessary to keep many consumption choices viable, especially when capital approaches the constraint-set boundaries.