In infinite horizon, heterogeneous-agent and incomplete-market models, the existence of an interior Ramsey steady state is often assumed instead of proven. This paper makes two fundamental contributions: (i) We prove that the interior Ramsey steady state assumed by Aiyagari (1995) does not exist in the standard Aiyagari model. Specifically, a steady state featuring the modified golden rule and a positive capital tax is feasible but not optimal. (ii) We design a modified, analytically tractable version of the standard Aiyagari model to unveil the necessary and/or sufficient conditions for the existence of a Ramsey steady state. These conditions are shown to be quite demanding and sensitive to structural parameter values pertaining to the economy's fiscal space for providing full self-insurance, such as the government's capacity to finance public debt, the degree of intertemporal elasticity of substitution, and the extent of history dependence of individual wealth on idiosyncratic shocks. In addition, we characterize the basic properties of both interior and non-interior Ramsey steady states and show that researchers may draw fundamentally misleading conclusions on optimal fiscal policies (such as the optimal capital tax rate) from their analysis when an interior Ramsey steady state is erroneously assumed to exist.