Several methods have been proposed to obtain stationarity in open economy models. I find substantial qualitative and quantitative differences between these methods in a two-country framework, in contrast to the results of Schmitt-Grohé and Uribe (2003). In models with a debt elastic interest rate premium or a convex portfolio cost, both the steady state and the equilibrium dynamics are unique if the elasticity of substitution between the domestic and the foreign traded good is high. However, there are three steady states if the elasticity of substitution is sufficiently low. With endogenous discounting, there is always a unique and stable steady state irrespective of the magnitude of the elasticity of substitution. Similar to the model with convex portfolio costs or a debt elastic interest rate premium, though, there can be multiple convergence paths for low values of the elasticity in response to shocks.